[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.50,0:00:06.56,Default,,0000,0000,0000,,In this video, we're going to be\Nlooking at sequences and series,
Dialogue: 0,0:00:06.56,0:00:11.10,Default,,0000,0000,0000,,so let's begin by looking at\Nwhat a sequences.
Dialogue: 0,0:00:12.03,0:00:15.88,Default,,0000,0000,0000,,This, for instance is a
Dialogue: 0,0:00:15.88,0:00:23.28,Default,,0000,0000,0000,,sequence. It's a\Nset of numbers.
Dialogue: 0,0:00:23.83,0:00:30.74,Default,,0000,0000,0000,,And here we seem to have a rule.\NAll of these are odd numbers, or
Dialogue: 0,0:00:30.74,0:00:36.28,Default,,0000,0000,0000,,we can look at it. We increase\Nby two each time 13579.
Dialogue: 0,0:00:37.28,0:00:42.46,Default,,0000,0000,0000,,So there's our sequence of\Nodd numbers.
Dialogue: 0,0:00:43.90,0:00:45.79,Default,,0000,0000,0000,,Here is another sequence.
Dialogue: 0,0:00:46.65,0:00:53.74,Default,,0000,0000,0000,,These numbers\Nare the
Dialogue: 0,0:00:53.74,0:01:01.10,Default,,0000,0000,0000,,square numbers.\N1 squared, 2 squared, is 4 three
Dialogue: 0,0:01:01.10,0:01:07.07,Default,,0000,0000,0000,,squared is 9. Four squared is 16\Nand 5 squared is 25. So again
Dialogue: 0,0:01:07.07,0:01:12.60,Default,,0000,0000,0000,,we've got a sequence of numbers.\NWe've got a rule that seems to
Dialogue: 0,0:01:12.60,0:01:15.27,Default,,0000,0000,0000,,produce them. Those are the.
Dialogue: 0,0:01:15.89,0:01:22.14,Default,,0000,0000,0000,,Square numbers. Is a\Nslightly different sequence.
Dialogue: 0,0:01:22.51,0:01:27.45,Default,,0000,0000,0000,,Here we've got alternation\Nbetween one and minus one back
Dialogue: 0,0:01:27.45,0:01:34.37,Default,,0000,0000,0000,,to one on again to minus one\Nback to one on again to minus
Dialogue: 0,0:01:34.37,0:01:38.81,Default,,0000,0000,0000,,one. But this is still a\Nsequence of numbers.
Dialogue: 0,0:01:39.51,0:01:42.97,Default,,0000,0000,0000,,Now, because I've written some\Ndots after it here.
Dialogue: 0,0:01:43.66,0:01:48.77,Default,,0000,0000,0000,,This means that this is meant to\Nbe an infinite sequence. It goes
Dialogue: 0,0:01:48.77,0:01:53.48,Default,,0000,0000,0000,,on forever and this is meant to\Nbe an infinite sequence. It
Dialogue: 0,0:01:53.48,0:01:58.59,Default,,0000,0000,0000,,carries on forever, and this one\Ndoes too. If I want a finite
Dialogue: 0,0:01:58.59,0:02:05.57,Default,,0000,0000,0000,,sequence. What might\Na finite sequence
Dialogue: 0,0:02:05.57,0:02:09.19,Default,,0000,0000,0000,,look like, for
Dialogue: 0,0:02:09.19,0:02:15.90,Default,,0000,0000,0000,,instance 1359? That would\Nbe a finite sequence. We've got
Dialogue: 0,0:02:15.90,0:02:18.65,Default,,0000,0000,0000,,4 numbers and then it stops
Dialogue: 0,0:02:18.65,0:02:25.16,Default,,0000,0000,0000,,dead. Perhaps if we look\Nat the sequence of square
Dialogue: 0,0:02:25.16,0:02:30.96,Default,,0000,0000,0000,,numbers 1, four, 916, that again\Nis a finite sequence.
Dialogue: 0,0:02:32.07,0:02:39.95,Default,,0000,0000,0000,,Sequence that will be very\Ninterested in is the sequence
Dialogue: 0,0:02:39.95,0:02:47.04,Default,,0000,0000,0000,,of whole numbers, the counting\Nnumbers, the integers. So
Dialogue: 0,0:02:47.04,0:02:54.92,Default,,0000,0000,0000,,there's a sequence of integers,\Nand it's finite because it
Dialogue: 0,0:02:54.92,0:03:02.80,Default,,0000,0000,0000,,stops at N, so we're\Ncounting 123456789 up to N.
Dialogue: 0,0:03:02.82,0:03:08.15,Default,,0000,0000,0000,,And the length of this sequence\Nis an the integer, the number N.
Dialogue: 0,0:03:10.31,0:03:16.28,Default,,0000,0000,0000,,Very popular sequence of\Nnumbers. Quite well known is
Dialogue: 0,0:03:16.28,0:03:18.27,Default,,0000,0000,0000,,this particular sequence.
Dialogue: 0,0:03:19.00,0:03:23.61,Default,,0000,0000,0000,,This is a slightly different\Nsequence. It's infinite keeps on
Dialogue: 0,0:03:23.61,0:03:25.92,Default,,0000,0000,0000,,going and it's called the
Dialogue: 0,0:03:25.92,0:03:30.71,Default,,0000,0000,0000,,Fibonacci sequence. And we can\Nsee how it's generated. This
Dialogue: 0,0:03:30.71,0:03:35.82,Default,,0000,0000,0000,,number 2 is formed by adding the\None and the one together, and
Dialogue: 0,0:03:35.82,0:03:40.54,Default,,0000,0000,0000,,then the three is formed by\Nadding the one and the two
Dialogue: 0,0:03:40.54,0:03:45.26,Default,,0000,0000,0000,,together. The Five is formed by\Nadding the two and the three
Dialogue: 0,0:03:45.26,0:03:46.83,Default,,0000,0000,0000,,together, and so on.
Dialogue: 0,0:03:47.84,0:03:52.58,Default,,0000,0000,0000,,So there's a question here. How\Ncould we write this rule down in
Dialogue: 0,0:03:52.58,0:03:56.60,Default,,0000,0000,0000,,general, where we can say it\Nthat any particular term is
Dialogue: 0,0:03:56.60,0:04:00.62,Default,,0000,0000,0000,,generated by adding the two\Nnumbers that come before it in
Dialogue: 0,0:04:00.62,0:04:04.63,Default,,0000,0000,0000,,the sequence together? But how\Nmight we set that down? How
Dialogue: 0,0:04:04.63,0:04:09.74,Default,,0000,0000,0000,,might we label it? One way might\Nbe to use algebra and say will
Dialogue: 0,0:04:09.74,0:04:14.12,Default,,0000,0000,0000,,call the first term you, and\Nbecause it's the first time we
Dialogue: 0,0:04:14.12,0:04:17.04,Default,,0000,0000,0000,,want to label it, so we call it
Dialogue: 0,0:04:17.04,0:04:20.69,Default,,0000,0000,0000,,you one. And then the next\Nterminal sequence, the second
Dialogue: 0,0:04:20.69,0:04:25.30,Default,,0000,0000,0000,,term. It would make sense to\Ncall it you too, and the third
Dialogue: 0,0:04:25.30,0:04:29.56,Default,,0000,0000,0000,,term in our sequence. It would\Nmake sense therefore, to call it
Dialogue: 0,0:04:29.56,0:04:36.92,Default,,0000,0000,0000,,you 3. You four and so on\Nup to UN. So this represents a
Dialogue: 0,0:04:36.92,0:04:43.01,Default,,0000,0000,0000,,finite sequence that's got N\Nterms in it. If we look at
Dialogue: 0,0:04:43.01,0:04:48.60,Default,,0000,0000,0000,,the Fibonacci sequence as an\Nexample of making use of this
Dialogue: 0,0:04:48.60,0:04:54.70,Default,,0000,0000,0000,,kind of notation, we could say\Nthat the end term UN was
Dialogue: 0,0:04:54.70,0:04:59.78,Default,,0000,0000,0000,,generated by adding together the\Ntwo terms that come immediately
Dialogue: 0,0:04:59.78,0:05:01.30,Default,,0000,0000,0000,,before it will.
Dialogue: 0,0:05:01.32,0:05:05.11,Default,,0000,0000,0000,,Term that comes immediately\Nbefore this must have a number
Dialogue: 0,0:05:05.11,0:05:10.42,Default,,0000,0000,0000,,attached to it. That's one less\Nthan N and that would be N minus
Dialogue: 0,0:05:10.42,0:05:15.91,Default,,0000,0000,0000,,one. Plus on the term that's\Ndown, the term that comes
Dialogue: 0,0:05:15.91,0:05:20.93,Default,,0000,0000,0000,,immediately before this one must\Nhave a number attached to it.
Dialogue: 0,0:05:20.93,0:05:26.85,Default,,0000,0000,0000,,That's one less than that. Well,\Nthat's UN minus 1 - 1 taking
Dialogue: 0,0:05:26.85,0:05:30.05,Default,,0000,0000,0000,,away 2 ones were taking away two
Dialogue: 0,0:05:30.05,0:05:36.64,Default,,0000,0000,0000,,altogether. So that we can see\Nhow we might use the algebra
Dialogue: 0,0:05:36.64,0:05:42.55,Default,,0000,0000,0000,,this algebraic notation help us\Nwrite down a rule for the Fibo
Dialogue: 0,0:05:42.55,0:05:46.41,Default,,0000,0000,0000,,Nachi sequence. OK, how can
Dialogue: 0,0:05:46.41,0:05:52.18,Default,,0000,0000,0000,,we? Use this in a\Nslightly different way.
Dialogue: 0,0:05:52.71,0:06:00.27,Default,,0000,0000,0000,,What we need to look at now\Nis to move on and have a
Dialogue: 0,0:06:00.27,0:06:04.05,Default,,0000,0000,0000,,look what we mean by a series.
Dialogue: 0,0:06:04.63,0:06:11.72,Default,,0000,0000,0000,,This is a\Nsequence, label it.
Dialogue: 0,0:06:12.49,0:06:17.33,Default,,0000,0000,0000,,A sequence it's a list of\Nnumbers generated by some
Dialogue: 0,0:06:17.33,0:06:22.17,Default,,0000,0000,0000,,particular rule. It's finite\Nbecause there are any of them.
Dialogue: 0,0:06:22.90,0:06:28.51,Default,,0000,0000,0000,,What then, is a series series is\Nwhat we get.
Dialogue: 0,0:06:29.28,0:06:32.97,Default,,0000,0000,0000,,When we add.
Dialogue: 0,0:06:32.97,0:06:36.48,Default,,0000,0000,0000,,Terms of the sequence.
Dialogue: 0,0:06:37.07,0:06:44.08,Default,,0000,0000,0000,,Together And because\Nwe're adding together and terms
Dialogue: 0,0:06:44.08,0:06:46.93,Default,,0000,0000,0000,,will call this SN.
Dialogue: 0,0:06:47.86,0:06:55.61,Default,,0000,0000,0000,,The sum of N terms\Nand it's that which is
Dialogue: 0,0:06:55.61,0:06:57.16,Default,,0000,0000,0000,,the series.
Dialogue: 0,0:06:59.04,0:07:06.10,Default,,0000,0000,0000,,So. Let's have a\Nlook at the sequence
Dialogue: 0,0:07:06.10,0:07:12.82,Default,,0000,0000,0000,,of numbers 123456, and\Nso on up to
Dialogue: 0,0:07:12.82,0:07:13.67,Default,,0000,0000,0000,,N.
Dialogue: 0,0:07:15.08,0:07:17.19,Default,,0000,0000,0000,,Then S1.
Dialogue: 0,0:07:18.85,0:07:22.21,Default,,0000,0000,0000,,Is just one.
Dialogue: 0,0:07:22.84,0:07:24.70,Default,,0000,0000,0000,,S2.
Dialogue: 0,0:07:25.75,0:07:32.66,Default,,0000,0000,0000,,Is the sum of the\Nfirst 2 terms 1 +
Dialogue: 0,0:07:32.66,0:07:34.98,Default,,0000,0000,0000,,2? And that gives us 3.
Dialogue: 0,0:07:35.74,0:07:42.89,Default,,0000,0000,0000,,S3. Is\Nthe sum of the first three
Dialogue: 0,0:07:42.89,0:07:46.61,Default,,0000,0000,0000,,terms 1 + 2 + 3?
Dialogue: 0,0:07:47.22,0:07:50.80,Default,,0000,0000,0000,,And that gives us 6
Dialogue: 0,0:07:50.80,0:07:58.25,Default,,0000,0000,0000,,and S4. Is the sum\Nof the first four terms 1 +
Dialogue: 0,0:07:58.25,0:08:02.10,Default,,0000,0000,0000,,2 + 3 + 4 and that
Dialogue: 0,0:08:02.10,0:08:03.58,Default,,0000,0000,0000,,gives us. Hey.
Dialogue: 0,0:08:04.48,0:08:10.15,Default,,0000,0000,0000,,So this gives us the basic\Nvocabulary to be able to move on
Dialogue: 0,0:08:10.15,0:08:14.94,Default,,0000,0000,0000,,to the next section of the\Nvideo, but just let's remind
Dialogue: 0,0:08:14.94,0:08:16.69,Default,,0000,0000,0000,,ourselves first of all.
Dialogue: 0,0:08:17.22,0:08:18.64,Default,,0000,0000,0000,,A sequence.
Dialogue: 0,0:08:20.18,0:08:24.20,Default,,0000,0000,0000,,Is a set of numbers\Ngenerated by some rule.
Dialogue: 0,0:08:25.83,0:08:31.40,Default,,0000,0000,0000,,A series is what we get when we\Nadd the terms of the sequence
Dialogue: 0,0:08:31.40,0:08:37.52,Default,,0000,0000,0000,,together. This particular\Nsequence has N terms in it
Dialogue: 0,0:08:37.52,0:08:43.49,Default,,0000,0000,0000,,because we've labeled each term\Nin the sequence with accounting
Dialogue: 0,0:08:43.49,0:08:49.46,Default,,0000,0000,0000,,number. If you like U1U2U free,\Nyou fall you N.
Dialogue: 0,0:08:51.38,0:08:55.92,Default,,0000,0000,0000,,Now. With this vocabulary of\Nsequences and series in mind,
Dialogue: 0,0:08:55.92,0:09:01.76,Default,,0000,0000,0000,,we're going to go on and have a\Nlook at a 2 special kinds of
Dialogue: 0,0:09:01.76,0:09:06.63,Default,,0000,0000,0000,,sequences. The first one is\Ncalled an arithmetic progression
Dialogue: 0,0:09:06.63,0:09:09.99,Default,,0000,0000,0000,,and the second one is called a
Dialogue: 0,0:09:09.99,0:09:15.12,Default,,0000,0000,0000,,geometric progression. Will\Nbegin with an arithmetic
Dialogue: 0,0:09:15.12,0:09:22.06,Default,,0000,0000,0000,,progression. Let's start by\Nhaving a look at this
Dialogue: 0,0:09:22.06,0:09:24.95,Default,,0000,0000,0000,,sequence of. Odd
Dialogue: 0,0:09:25.47,0:09:32.75,Default,,0000,0000,0000,,Numbers that we\Nhad before 1357.
Dialogue: 0,0:09:33.30,0:09:40.03,Default,,0000,0000,0000,,Is another sequence\Nnot 1020 thirty,
Dialogue: 0,0:09:40.03,0:09:43.39,Default,,0000,0000,0000,,and so on.
Dialogue: 0,0:09:44.64,0:09:51.63,Default,,0000,0000,0000,,What we can see in this first\Nsequence is that each term after
Dialogue: 0,0:09:51.63,0:09:56.48,Default,,0000,0000,0000,,the first one is formed by\Nadding on to.
Dialogue: 0,0:09:57.21,0:10:00.26,Default,,0000,0000,0000,,1 + 2 gives us 3.
Dialogue: 0,0:10:00.84,0:10:03.86,Default,,0000,0000,0000,,3 + 2 gives us 5.
Dialogue: 0,0:10:04.71,0:10:11.14,Default,,0000,0000,0000,,5 + 2 gives us 7 and it's\Nbecause we're adding on the
Dialogue: 0,0:10:11.14,0:10:17.08,Default,,0000,0000,0000,,same amount every time. This\Nis an example of what we call
Dialogue: 0,0:10:17.08,0:10:18.08,Default,,0000,0000,0000,,an arithmetic.
Dialogue: 0,0:10:19.87,0:10:28.08,Default,,0000,0000,0000,,Progression.\NIf we look at this sequence of
Dialogue: 0,0:10:28.08,0:10:33.54,Default,,0000,0000,0000,,numbers, we can see exactly the\Nsame property we've started with
Dialogue: 0,0:10:33.54,0:10:39.99,Default,,0000,0000,0000,,zero. We've added on 10, and\Nwe've added on 10 again to get
Dialogue: 0,0:10:39.99,0:10:46.93,Default,,0000,0000,0000,,20. We've had it on 10 again to\Nget 30, so again, this is
Dialogue: 0,0:10:46.93,0:10:49.41,Default,,0000,0000,0000,,exactly the same. It's an
Dialogue: 0,0:10:49.41,0:10:56.17,Default,,0000,0000,0000,,arithmetic progression. We don't\Nhave to add on things, so
Dialogue: 0,0:10:56.17,0:11:03.28,Default,,0000,0000,0000,,for instance a sequence of\Nnumbers that went like this 8
Dialogue: 0,0:11:03.28,0:11:05.87,Default,,0000,0000,0000,,five, 2 - 1.
Dialogue: 0,0:11:06.56,0:11:08.66,Default,,0000,0000,0000,,Minus
Dialogue: 0,0:11:08.66,0:11:14.25,Default,,0000,0000,0000,,4. If we look what's\Nhappening where going from 8:00
Dialogue: 0,0:11:14.25,0:11:18.51,Default,,0000,0000,0000,,to 5:00, so that's takeaway\Nthree were going from five to
Dialogue: 0,0:11:18.51,0:11:23.15,Default,,0000,0000,0000,,two, so that's takeaway. Three\Nwere going from 2 to minus. One
Dialogue: 0,0:11:23.15,0:11:27.41,Default,,0000,0000,0000,,takeaway. Three were going from\Nminus one to minus four takeaway
Dialogue: 0,0:11:27.41,0:11:33.66,Default,,0000,0000,0000,,3. Another way of thinking about\Ntakeaway three is to say where
Dialogue: 0,0:11:33.66,0:11:35.51,Default,,0000,0000,0000,,adding on minus three.
Dialogue: 0,0:11:36.33,0:11:43.05,Default,,0000,0000,0000,,8 at minus three is 5 five at\Nminus three is 2, two AD minus
Dialogue: 0,0:11:43.05,0:11:48.43,Default,,0000,0000,0000,,three is minus one, so again,\Nthis is an example of an
Dialogue: 0,0:11:48.43,0:11:54.08,Default,,0000,0000,0000,,arithmetic progression. And what\Nwe want to be able to do is to
Dialogue: 0,0:11:54.08,0:11:56.100,Default,,0000,0000,0000,,try and encapsulate this\Narithmetic progression in some
Dialogue: 0,0:11:56.100,0:11:59.18,Default,,0000,0000,0000,,algebra, so we'll use the letter
Dialogue: 0,0:11:59.18,0:12:04.69,Default,,0000,0000,0000,,A. To stand for\Nthe first term.
Dialogue: 0,0:12:05.31,0:12:12.36,Default,,0000,0000,0000,,And will use the letter\ND to stand for the
Dialogue: 0,0:12:12.36,0:12:17.77,Default,,0000,0000,0000,,common difference. Now the\Ncommon difference is the
Dialogue: 0,0:12:17.77,0:12:23.69,Default,,0000,0000,0000,,difference between each term and\Nit's called common because it is
Dialogue: 0,0:12:23.69,0:12:30.14,Default,,0000,0000,0000,,common to each between each\Nterm. So let's have a look at
Dialogue: 0,0:12:30.14,0:12:36.60,Default,,0000,0000,0000,,one 357 and let's have a think\Nabout how it's structured 13.
Dialogue: 0,0:12:37.34,0:12:44.82,Default,,0000,0000,0000,,5. 7 and so\Non. So we begin with one and
Dialogue: 0,0:12:44.82,0:12:48.82,Default,,0000,0000,0000,,then the three is 1 + 2.
Dialogue: 0,0:12:50.27,0:12:57.76,Default,,0000,0000,0000,,The Five is 1 + 2 tools because\Nby the time we got to five,
Dialogue: 0,0:12:57.76,0:13:00.75,Default,,0000,0000,0000,,we've added four onto the one.
Dialogue: 0,0:13:01.42,0:13:07.89,Default,,0000,0000,0000,,The Seven is one plus. Now the\Ntime we've got to Seven, we've
Dialogue: 0,0:13:07.89,0:13:14.37,Default,,0000,0000,0000,,added three tools on. Let's just\Ndo one more. Let's put nine in
Dialogue: 0,0:13:14.37,0:13:19.85,Default,,0000,0000,0000,,there and that would be 1 + 4\Ntimes by two.
Dialogue: 0,0:13:20.98,0:13:25.32,Default,,0000,0000,0000,,So let's see if we can begin to\Nwrite this down. This is one.
Dialogue: 0,0:13:26.28,0:13:33.24,Default,,0000,0000,0000,,Now what have we got here? This\Nis the second term in the
Dialogue: 0,0:13:33.24,0:13:39.77,Default,,0000,0000,0000,,series. But we've only got 1 two\Nthere, so if you like we've got
Dialogue: 0,0:13:39.77,0:13:43.14,Default,,0000,0000,0000,,1 + 2 - 1 times by two.
Dialogue: 0,0:13:44.06,0:13:48.78,Default,,0000,0000,0000,,One plus now, what's multiplying\Nthe two here? Well, this is the
Dialogue: 0,0:13:48.78,0:13:50.74,Default,,0000,0000,0000,,third term in the series.
Dialogue: 0,0:13:51.65,0:13:59.31,Default,,0000,0000,0000,,So we've got a 2 here,\Nso we're multiplying by 3 -
Dialogue: 0,0:13:59.31,0:14:05.20,Default,,0000,0000,0000,,1. Here this is term\N#4 and we're
Dialogue: 0,0:14:05.20,0:14:10.93,Default,,0000,0000,0000,,multiplying by three,\Nso that's 4 - 1 times
Dialogue: 0,0:14:10.93,0:14:17.30,Default,,0000,0000,0000,,by two. And here this\Nis term #5, so we've
Dialogue: 0,0:14:17.30,0:14:23.04,Default,,0000,0000,0000,,got 1 + 5 - 1\Ntimes by two.
Dialogue: 0,0:14:24.43,0:14:28.67,Default,,0000,0000,0000,,Now, if we think about\Nwhat's happening here.
Dialogue: 0,0:14:31.12,0:14:33.27,Default,,0000,0000,0000,,We're starting with A.
Dialogue: 0,0:14:34.67,0:14:37.61,Default,,0000,0000,0000,,And then on to the A. We're
Dialogue: 0,0:14:37.61,0:14:44.94,Default,,0000,0000,0000,,adding D. Then we're adding on\Nanother day, so that's a plus
Dialogue: 0,0:14:44.94,0:14:51.83,Default,,0000,0000,0000,,2D, and then we're adding on\Nanother D. So that's a plus
Dialogue: 0,0:14:51.83,0:14:57.21,Default,,0000,0000,0000,,3D. The question is, if we've\Ngot N terms in our sequence,
Dialogue: 0,0:14:57.21,0:15:02.45,Default,,0000,0000,0000,,then what's the last term? But\Nif we look, we can see that the
Dialogue: 0,0:15:02.45,0:15:04.32,Default,,0000,0000,0000,,first term was just a.
Dialogue: 0,0:15:04.88,0:15:12.24,Default,,0000,0000,0000,,The second term was a plus, one\ND. The third term was a plus
Dialogue: 0,0:15:12.24,0:15:19.61,Default,,0000,0000,0000,,2D. The fourth term was a plus\N3D, so the end term must be
Dialogue: 0,0:15:19.61,0:15:22.24,Default,,0000,0000,0000,,a plus N minus one.
Dialogue: 0,0:15:22.75,0:15:23.23,Default,,0000,0000,0000,,Gay.
Dialogue: 0,0:15:25.20,0:15:32.15,Default,,0000,0000,0000,,Now, this last term of\Nour sequence, we often label
Dialogue: 0,0:15:32.15,0:15:35.62,Default,,0000,0000,0000,,L and call it the
Dialogue: 0,0:15:35.62,0:15:37.02,Default,,0000,0000,0000,,last term.
Dialogue: 0,0:15:37.07,0:15:39.65,Default,,0000,0000,0000,,Or
Dialogue: 0,0:15:40.42,0:15:42.53,Default,,0000,0000,0000,,The end.
Dialogue: 0,0:15:43.14,0:15:46.75,Default,,0000,0000,0000,,Turn. To be more mathematical
Dialogue: 0,0:15:46.75,0:15:51.37,Default,,0000,0000,0000,,about it. And one of the things\Nthat we'd like to be able to do
Dialogue: 0,0:15:51.37,0:15:54.57,Default,,0000,0000,0000,,with a sequence of numbers like\Nthis is get to a series. In
Dialogue: 0,0:15:54.57,0:15:58.26,Default,,0000,0000,0000,,other words, to be able to add\Nthem up. So let's have a look at
Dialogue: 0,0:15:58.26,0:16:05.69,Default,,0000,0000,0000,,that. So SN the some of these\Nend terms is A plus A+B plus
Dialogue: 0,0:16:05.69,0:16:12.85,Default,,0000,0000,0000,,A plus 2B plus. But I want\Njust to stop there and what I
Dialogue: 0,0:16:12.85,0:16:19.49,Default,,0000,0000,0000,,want to do is I want to\Nstart at the end. This end
Dialogue: 0,0:16:19.49,0:16:24.09,Default,,0000,0000,0000,,now now the last one will be\Nplus L.
Dialogue: 0,0:16:25.15,0:16:29.52,Default,,0000,0000,0000,,So what will be the next one\Nback when we generate each term
Dialogue: 0,0:16:29.52,0:16:35.23,Default,,0000,0000,0000,,by adding on D. So we added on D\Nto this one to get L. So this
Dialogue: 0,0:16:35.23,0:16:37.58,Default,,0000,0000,0000,,one's got to be L minus D.
Dialogue: 0,0:16:38.92,0:16:45.75,Default,,0000,0000,0000,,And the one before that\None similarly will be L
Dialogue: 0,0:16:45.75,0:16:51.52,Default,,0000,0000,0000,,minus 2D. On the rest of the\Nterms will be in between.
Dialogue: 0,0:16:52.64,0:16:54.100,Default,,0000,0000,0000,,Now I'm going to use a trick.
Dialogue: 0,0:16:55.54,0:16:58.85,Default,,0000,0000,0000,,Mathematicians often use. I'm\Ngoing to write this down the
Dialogue: 0,0:16:58.85,0:17:02.07,Default,,0000,0000,0000,,other way around. So I have L
Dialogue: 0,0:17:02.07,0:17:05.87,Default,,0000,0000,0000,,there. Plus L minus
Dialogue: 0,0:17:05.87,0:17:12.43,Default,,0000,0000,0000,,D. Plus L minus 2D plus\Nplus. Now what will I have?
Dialogue: 0,0:17:12.43,0:17:17.87,Default,,0000,0000,0000,,Well, writing this down either\Nway around, I'll Have A at the
Dialogue: 0,0:17:17.87,0:17:21.26,Default,,0000,0000,0000,,end. Then I'll have this next
Dialogue: 0,0:17:21.26,0:17:23.73,Default,,0000,0000,0000,,term a. Plus D.
Dialogue: 0,0:17:24.23,0:17:28.97,Default,,0000,0000,0000,,And I'll have this next\Nterm, A plus 2D.
Dialogue: 0,0:17:31.49,0:17:36.32,Default,,0000,0000,0000,,Now I'm going to add these two\Ntogether. Let's look what
Dialogue: 0,0:17:36.32,0:17:41.59,Default,,0000,0000,0000,,happens if I add SN&SN together.\NI've just got two of them.
Dialogue: 0,0:17:42.97,0:17:49.55,Default,,0000,0000,0000,,By ad A&L together I get a\Nplus L let me just group
Dialogue: 0,0:17:49.55,0:17:50.56,Default,,0000,0000,0000,,those together.
Dialogue: 0,0:17:51.82,0:17:59.00,Default,,0000,0000,0000,,Now I've got a plus D&L Minus D,\Nso if I add them together I have
Dialogue: 0,0:17:59.00,0:18:06.19,Default,,0000,0000,0000,,a plus L Plus D minus D, so\Nall I've got left is A plus L.
Dialogue: 0,0:18:07.29,0:18:14.19,Default,,0000,0000,0000,,But the same thing is going to\Nhappen here. I have a plus L
Dialogue: 0,0:18:14.19,0:18:19.12,Default,,0000,0000,0000,,Plus 2D Takeaway 2D, so again\Njust a plus L.
Dialogue: 0,0:18:20.03,0:18:25.07,Default,,0000,0000,0000,,When we get down To this end,\Nit's still the same thing
Dialogue: 0,0:18:25.07,0:18:30.11,Default,,0000,0000,0000,,happening. I've A plus L\Ntakeaway 2D add onto D so again
Dialogue: 0,0:18:30.11,0:18:35.57,Default,,0000,0000,0000,,the DS have disappeared. If you\Nlike and I've got L plus A.
Dialogue: 0,0:18:36.48,0:18:43.88,Default,,0000,0000,0000,,Plus a plusle takeaway D add\Non DLA and right at the
Dialogue: 0,0:18:43.88,0:18:46.97,Default,,0000,0000,0000,,end. L plus a again.
Dialogue: 0,0:18:48.64,0:18:53.33,Default,,0000,0000,0000,,Well, how many of these have I\Ngot? But I've got N terms.
Dialogue: 0,0:18:54.13,0:19:01.73,Default,,0000,0000,0000,,In each of these lines of sums,\Nso I must still have end terms
Dialogue: 0,0:19:01.73,0:19:07.70,Default,,0000,0000,0000,,here, and so this must be an\Ntimes a plus L.
Dialogue: 0,0:19:08.39,0:19:15.53,Default,,0000,0000,0000,,And so if we now divide\Nboth sides by two, we have.
Dialogue: 0,0:19:15.53,0:19:22.67,Default,,0000,0000,0000,,SN is 1/2 of N times\Nby a plus L and that
Dialogue: 0,0:19:22.67,0:19:28.62,Default,,0000,0000,0000,,gives us our some of the\Nterms of an arithmetic
Dialogue: 0,0:19:28.62,0:19:35.41,Default,,0000,0000,0000,,progression. Let's just write\Ndown again the two results that
Dialogue: 0,0:19:35.41,0:19:42.74,Default,,0000,0000,0000,,we've got. We've got L the\Nend term, or the final term
Dialogue: 0,0:19:42.74,0:19:50.07,Default,,0000,0000,0000,,is equal to a plus N\Nminus one times by D and
Dialogue: 0,0:19:50.07,0:19:53.74,Default,,0000,0000,0000,,we've got the SN is 1/2.
Dialogue: 0,0:19:54.32,0:20:01.38,Default,,0000,0000,0000,,Times by N number of\Nterms times by a plus
Dialogue: 0,0:20:01.38,0:20:02.09,Default,,0000,0000,0000,,L.
Dialogue: 0,0:20:03.29,0:20:09.06,Default,,0000,0000,0000,,Now, one thing we can do is take\Nthis expression for L and
Dialogue: 0,0:20:09.06,0:20:10.84,Default,,0000,0000,0000,,substitute it into here.
Dialogue: 0,0:20:11.77,0:20:19.57,Default,,0000,0000,0000,,Replacing this al, so let's do\Nthat. SN is equal to 1/2.
Dialogue: 0,0:20:20.18,0:20:27.72,Default,,0000,0000,0000,,Times by N number of terms\Ntimes by a plus and instead
Dialogue: 0,0:20:27.72,0:20:35.25,Default,,0000,0000,0000,,of L will write this a\Nplus N minus one times by
Dialogue: 0,0:20:35.25,0:20:39.56,Default,,0000,0000,0000,,D. April say gives us\Ntwo way.
Dialogue: 0,0:20:40.60,0:20:47.64,Default,,0000,0000,0000,,So the sum of the\Nend terms is 1/2 an
Dialogue: 0,0:20:47.64,0:20:51.16,Default,,0000,0000,0000,,2A plus N minus 1D.
Dialogue: 0,0:20:51.93,0:20:53.96,Default,,0000,0000,0000,,Close the bracket.
Dialogue: 0,0:20:55.73,0:21:02.06,Default,,0000,0000,0000,,And these. That I'm\Nunderlining are the three
Dialogue: 0,0:21:02.06,0:21:05.84,Default,,0000,0000,0000,,important things about an\Narithmetic progression.
Dialogue: 0,0:21:07.28,0:21:11.07,Default,,0000,0000,0000,,If A is the first
Dialogue: 0,0:21:11.07,0:21:16.83,Default,,0000,0000,0000,,term. And D is\Nthe common difference.
Dialogue: 0,0:21:17.75,0:21:23.10,Default,,0000,0000,0000,,And N\Nis the
Dialogue: 0,0:21:23.10,0:21:27.12,Default,,0000,0000,0000,,number of\Nterms.
Dialogue: 0,0:21:28.37,0:21:33.54,Default,,0000,0000,0000,,In our arithmetic progression,\Nthen, this expression gives us
Dialogue: 0,0:21:33.54,0:21:39.87,Default,,0000,0000,0000,,the NTH or the last term.\NThis expression gives us the
Dialogue: 0,0:21:39.87,0:21:46.20,Default,,0000,0000,0000,,some of those N terms, and\Nthis expression gives us also
Dialogue: 0,0:21:46.20,0:21:49.64,Default,,0000,0000,0000,,the sum of the end terms.
Dialogue: 0,0:21:50.56,0:21:54.52,Default,,0000,0000,0000,,One of the things that you also\Nneed to understand is that
Dialogue: 0,0:21:54.52,0:21:58.15,Default,,0000,0000,0000,,sometimes we like to shorten the\Nlanguage as well as using
Dialogue: 0,0:21:58.15,0:22:02.86,Default,,0000,0000,0000,,algebra. So that rather than\Nkeep saying arithmetic
Dialogue: 0,0:22:02.86,0:22:07.84,Default,,0000,0000,0000,,progression, we often refer to\Nthese as a peas.
Dialogue: 0,0:22:09.13,0:22:11.28,Default,,0000,0000,0000,,Now we've got some facts, some
Dialogue: 0,0:22:11.28,0:22:16.53,Default,,0000,0000,0000,,information there. So let's have\Na look at trying to see if we
Dialogue: 0,0:22:16.53,0:22:18.66,Default,,0000,0000,0000,,can use them to solve some
Dialogue: 0,0:22:18.66,0:22:25.80,Default,,0000,0000,0000,,questions. So let's have\Na look at this
Dialogue: 0,0:22:25.80,0:22:31.75,Default,,0000,0000,0000,,sequence of numbers again,\Nwhich we've identified.
Dialogue: 0,0:22:33.46,0:22:37.02,Default,,0000,0000,0000,,And let's ask ourselves\Nwhat's the sum?
Dialogue: 0,0:22:38.15,0:22:41.10,Default,,0000,0000,0000,,Of. The first
Dialogue: 0,0:22:41.78,0:22:48.28,Default,,0000,0000,0000,,50 terms So\Nwe could start to try and add
Dialogue: 0,0:22:48.28,0:22:54.37,Default,,0000,0000,0000,,them up. 1 + 3 is four and four\Nand five is 9, and nine and
Dialogue: 0,0:22:54.37,0:22:59.71,Default,,0000,0000,0000,,Seven is 16 and 16 and 9025, and\Nthen the next get or getting
Dialogue: 0,0:22:59.71,0:23:03.90,Default,,0000,0000,0000,,rather complicated. But we can\Nwrite down some facts about this
Dialogue: 0,0:23:03.90,0:23:08.47,Default,,0000,0000,0000,,straight away. We can write down\Nthat the first term is one.
Dialogue: 0,0:23:09.07,0:23:14.35,Default,,0000,0000,0000,,We can write down that the\Ncommon difference Dean is 2 and
Dialogue: 0,0:23:14.35,0:23:20.07,Default,,0000,0000,0000,,we can write down the number of\Nterms we're dealing with. An is
Dialogue: 0,0:23:20.07,0:23:27.04,Default,,0000,0000,0000,,50. We know we have a\Nformula that says SN is 1/2
Dialogue: 0,0:23:27.04,0:23:29.70,Default,,0000,0000,0000,,times the number of terms.
Dialogue: 0,0:23:30.71,0:23:37.70,Default,,0000,0000,0000,,Times 2A plus N minus 1D. So\Ninstead of having to add this up
Dialogue: 0,0:23:37.70,0:23:43.68,Default,,0000,0000,0000,,as though it was a big\Narithmetic sum a big problem, we
Dialogue: 0,0:23:43.68,0:23:48.67,Default,,0000,0000,0000,,can simply substitute the\Nnumbers into the formula. So SNS
Dialogue: 0,0:23:48.67,0:23:52.67,Default,,0000,0000,0000,,50 in this case is equal to 1/2.
Dialogue: 0,0:23:53.08,0:23:55.29,Default,,0000,0000,0000,,Times by 50.
Dialogue: 0,0:23:55.80,0:24:02.94,Default,,0000,0000,0000,,Times by two A That's just two\N2 * 1 plus N minus one
Dialogue: 0,0:24:02.94,0:24:06.51,Default,,0000,0000,0000,,and is 50, so N minus one
Dialogue: 0,0:24:06.51,0:24:10.21,Default,,0000,0000,0000,,is 49. Times by the common
Dialogue: 0,0:24:10.21,0:24:11.27,Default,,0000,0000,0000,,difference too.
Dialogue: 0,0:24:12.26,0:24:19.72,Default,,0000,0000,0000,,So. We\Ncan cancel a 2 into the 50
Dialogue: 0,0:24:19.72,0:24:23.23,Default,,0000,0000,0000,,that gives us 25 times by now.
Dialogue: 0,0:24:23.77,0:24:30.84,Default,,0000,0000,0000,,2 * 49 or 2 * 49\Nis 98 and two is 100, so
Dialogue: 0,0:24:30.84,0:24:37.40,Default,,0000,0000,0000,,we have 25 times by 100, so\Nthat's 2500. So what was going
Dialogue: 0,0:24:37.40,0:24:42.46,Default,,0000,0000,0000,,to be quite a lengthy and\Ndifficult calculation's come out
Dialogue: 0,0:24:42.46,0:24:48.70,Default,,0000,0000,0000,,quite quickly. Let's see if we\Ncan solve a more difficult
Dialogue: 0,0:24:48.70,0:24:53.00,Default,,0000,0000,0000,,problem.\N1.
Dialogue: 0,0:24:54.11,0:24:58.84,Default,,0000,0000,0000,,Plus 3.5.\N+6.
Dialogue: 0,0:25:00.05,0:25:03.62,Default,,0000,0000,0000,,Plus 8.5. Plus
Dialogue: 0,0:25:04.49,0:25:08.27,Default,,0000,0000,0000,,Plus 101.
Dialogue: 0,0:25:10.29,0:25:11.31,Default,,0000,0000,0000,,Add this up.
Dialogue: 0,0:25:12.49,0:25:19.55,Default,,0000,0000,0000,,Well. Can we identify what\Nkind of a series this is? We can
Dialogue: 0,0:25:19.55,0:25:25.16,Default,,0000,0000,0000,,see quite clearly that one to\N3.5 while that's a gap of 2.5
Dialogue: 0,0:25:25.16,0:25:32.05,Default,,0000,0000,0000,,and then a gap of 2.5 to 6. So\Nwhat we've got here is in fact
Dialogue: 0,0:25:32.05,0:25:36.79,Default,,0000,0000,0000,,an arithmetic progression, and\Nwe can see here. We've got 100
Dialogue: 0,0:25:36.79,0:25:42.83,Default,,0000,0000,0000,,and one at the end. Our last\Nterm is 101 and the first term
Dialogue: 0,0:25:42.83,0:25:45.84,Default,,0000,0000,0000,,is one. Now we know a formula.
Dialogue: 0,0:25:45.89,0:25:49.39,Default,,0000,0000,0000,,For the last term L.
Dialogue: 0,0:25:50.05,0:25:57.13,Default,,0000,0000,0000,,Equals A plus N minus\None times by D.
Dialogue: 0,0:25:58.35,0:26:05.13,Default,,0000,0000,0000,,Might just have a look at what\Nwe know in this formula. What we
Dialogue: 0,0:26:05.13,0:26:07.06,Default,,0000,0000,0000,,know L it's 101.
Dialogue: 0,0:26:07.07,0:26:12.23,Default,,0000,0000,0000,,We know a It's the first\Nterm, it's one.
Dialogue: 0,0:26:13.28,0:26:19.26,Default,,0000,0000,0000,,Plus Well, we have no idea what\Nany is. We don't know how many
Dialogue: 0,0:26:19.26,0:26:24.06,Default,,0000,0000,0000,,terms we've got, so that's N\Nminus one times by D and we know
Dialogue: 0,0:26:24.06,0:26:25.78,Default,,0000,0000,0000,,what that is, that's 2.5.
Dialogue: 0,0:26:26.51,0:26:31.79,Default,,0000,0000,0000,,Well, this is nothing more than\Nan equation for an, so let's
Dialogue: 0,0:26:31.79,0:26:37.51,Default,,0000,0000,0000,,begin by taking one from each\Nside. That gives us 100 equals N
Dialogue: 0,0:26:37.51,0:26:43.23,Default,,0000,0000,0000,,minus one times by 2.5. And now\NI'm going to divide both sides
Dialogue: 0,0:26:43.23,0:26:49.83,Default,,0000,0000,0000,,by 2.5 and that will give me 40\Nequals N minus one, and now I'll
Dialogue: 0,0:26:49.83,0:26:57.31,Default,,0000,0000,0000,,add 1 to both sides and so 41 is\Nequal to end, so I know how many
Dialogue: 0,0:26:57.31,0:27:04.20,Default,,0000,0000,0000,,terms that. Are in this series,\NSo what I can do now is I
Dialogue: 0,0:27:04.20,0:27:10.19,Default,,0000,0000,0000,,can add it up because the sum of\NN terms is 1/2.
Dialogue: 0,0:27:10.86,0:27:13.87,Default,,0000,0000,0000,,NA plus
Dialogue: 0,0:27:13.87,0:27:20.80,Default,,0000,0000,0000,,L. And I\Nnow know all these terms
Dialogue: 0,0:27:20.80,0:27:24.30,Default,,0000,0000,0000,,here have 1/2 * 41
Dialogue: 0,0:27:24.30,0:27:27.48,Default,,0000,0000,0000,,* 1. Plus
Dialogue: 0,0:27:27.48,0:27:34.79,Default,,0000,0000,0000,,101. Let me just turn\Nthe page over and write this
Dialogue: 0,0:27:34.79,0:27:36.46,Default,,0000,0000,0000,,some down again.
Dialogue: 0,0:27:37.12,0:27:43.56,Default,,0000,0000,0000,,SN is equal\Nto 1/2 *
Dialogue: 0,0:27:43.56,0:27:48.93,Default,,0000,0000,0000,,41 * 1\N+ 101.
Dialogue: 0,0:27:50.04,0:27:57.43,Default,,0000,0000,0000,,So we have 1/2 times by 41\Ntimes by 102 and we can cancel
Dialogue: 0,0:27:57.43,0:28:04.82,Default,,0000,0000,0000,,it to there to give US 41\Ntimes by 51. And to do that
Dialogue: 0,0:28:04.82,0:28:10.63,Default,,0000,0000,0000,,I'd want to get out my\NCalculator, but we'll leave it
Dialogue: 0,0:28:10.63,0:28:12.74,Default,,0000,0000,0000,,there to be finished.
Dialogue: 0,0:28:13.27,0:28:16.86,Default,,0000,0000,0000,,So that's one kind of problem.
Dialogue: 0,0:28:17.80,0:28:21.88,Default,,0000,0000,0000,,Let's have a look at another\Nkind of problem.
Dialogue: 0,0:28:22.43,0:28:28.21,Default,,0000,0000,0000,,Let's say we've got an\Narithmetic progression whose
Dialogue: 0,0:28:28.21,0:28:31.11,Default,,0000,0000,0000,,first term is 3.
Dialogue: 0,0:28:32.17,0:28:35.53,Default,,0000,0000,0000,,And the sum.
Dialogue: 0,0:28:36.20,0:28:37.57,Default,,0000,0000,0000,,Of.
Dialogue: 0,0:28:39.21,0:28:42.97,Default,,0000,0000,0000,,The first 8.
Dialogue: 0,0:28:44.74,0:28:45.63,Default,,0000,0000,0000,,Terms.
Dialogue: 0,0:28:47.02,0:28:53.97,Default,,0000,0000,0000,,Is twice.\NThe sum
Dialogue: 0,0:28:53.97,0:28:59.74,Default,,0000,0000,0000,,of the\Nfirst 5
Dialogue: 0,0:28:59.74,0:29:01.18,Default,,0000,0000,0000,,terms.
Dialogue: 0,0:29:02.57,0:29:04.82,Default,,0000,0000,0000,,And that seems really quite
Dialogue: 0,0:29:04.82,0:29:09.23,Default,,0000,0000,0000,,complicated. But it needn't\Nbe, but remember this is the
Dialogue: 0,0:29:09.23,0:29:10.37,Default,,0000,0000,0000,,same arithmetic progression.
Dialogue: 0,0:29:12.25,0:29:18.02,Default,,0000,0000,0000,,So let's have a think what this\Nis telling us A is equal to
Dialogue: 0,0:29:18.02,0:29:23.37,Default,,0000,0000,0000,,three and the sum of the first 8\Nterms. Well, to begin with,
Dialogue: 0,0:29:23.37,0:29:28.32,Default,,0000,0000,0000,,let's write down what the sum of\Nthe first 8 terms is.
Dialogue: 0,0:29:28.87,0:29:31.82,Default,,0000,0000,0000,,Well, it's a half.
Dialogue: 0,0:29:32.44,0:29:39.09,Default,,0000,0000,0000,,Times N Times\N2A plus and
Dialogue: 0,0:29:39.09,0:29:41.30,Default,,0000,0000,0000,,minus 1D.
Dialogue: 0,0:29:42.47,0:29:45.85,Default,,0000,0000,0000,,And N is equal to 8.
Dialogue: 0,0:29:46.93,0:29:48.75,Default,,0000,0000,0000,,So we've got a half.
Dialogue: 0,0:29:49.98,0:29:57.63,Default,,0000,0000,0000,,Times 8. 2A\Nplus N minus one is
Dialogue: 0,0:29:57.63,0:30:05.48,Default,,0000,0000,0000,,7D. So S 8\Nis equal to half of
Dialogue: 0,0:30:05.48,0:30:11.83,Default,,0000,0000,0000,,eight is 4 * 2\NA Plus 7D.
Dialogue: 0,0:30:12.97,0:30:20.43,Default,,0000,0000,0000,,But we also know that a\Nis equal to three, so we
Dialogue: 0,0:30:20.43,0:30:27.90,Default,,0000,0000,0000,,can put that in there as\Nwell. That's 4 * 6 because
Dialogue: 0,0:30:27.90,0:30:31.63,Default,,0000,0000,0000,,a is 3 + 7 D.
Dialogue: 0,0:30:32.34,0:30:39.81,Default,,0000,0000,0000,,Next one, the sum\Nof the first 5
Dialogue: 0,0:30:39.81,0:30:46.83,Default,,0000,0000,0000,,terms. Let me just write\Ndown some of the first
Dialogue: 0,0:30:46.83,0:30:48.86,Default,,0000,0000,0000,,8 terms were.
Dialogue: 0,0:30:49.15,0:30:55.55,Default,,0000,0000,0000,,4. Times\N6 minus
Dialogue: 0,0:30:55.55,0:31:01.15,Default,,0000,0000,0000,,plus 7D\Nfirst 5
Dialogue: 0,0:31:01.15,0:31:08.71,Default,,0000,0000,0000,,terms. Half times the number of\Nterms. That's 5 * 2 A plus
Dialogue: 0,0:31:08.71,0:31:15.89,Default,,0000,0000,0000,,N minus one times by D will.\NThat must be 4 because any is
Dialogue: 0,0:31:15.89,0:31:17.94,Default,,0000,0000,0000,,5 times by D.
Dialogue: 0,0:31:18.90,0:31:25.48,Default,,0000,0000,0000,,So much is 5 over 2 and\Nlet's remember that a is equal
Dialogue: 0,0:31:25.48,0:31:32.56,Default,,0000,0000,0000,,to three, so that 6 + 4\ND. So I've got S 8 and
Dialogue: 0,0:31:32.56,0:31:39.14,Default,,0000,0000,0000,,I've got S5 and the question\Nsaid that S8 was equal to twice
Dialogue: 0,0:31:39.14,0:31:42.97,Default,,0000,0000,0000,,as five. So I can write this
Dialogue: 0,0:31:42.97,0:31:48.64,Default,,0000,0000,0000,,for S8. Is\Nequal to
Dialogue: 0,0:31:48.64,0:31:56.03,Default,,0000,0000,0000,,twice. This which is\NS five 2 * 5 over two
Dialogue: 0,0:31:56.03,0:32:02.31,Default,,0000,0000,0000,,6 + 4 D and what seemed\Na very difficult question as
Dialogue: 0,0:32:02.31,0:32:08.06,Default,,0000,0000,0000,,reduced itself to an ordinary\Nlinear equation in terms of D.
Dialogue: 0,0:32:08.06,0:32:14.34,Default,,0000,0000,0000,,So we can do some cancelling\Nthere and we can multiply out
Dialogue: 0,0:32:14.34,0:32:21.66,Default,,0000,0000,0000,,the brackets for six is a 24\N+ 28, D is equal to 56R.
Dialogue: 0,0:32:21.69,0:32:24.97,Default,,0000,0000,0000,,30 + 5 fours
Dialogue: 0,0:32:24.97,0:32:31.87,Default,,0000,0000,0000,,are 20D. I can take\N20D from each side that gives me
Dialogue: 0,0:32:31.87,0:32:33.26,Default,,0000,0000,0000,,8 D there.
Dialogue: 0,0:32:33.83,0:32:40.100,Default,,0000,0000,0000,,And I can take 24 from each\Nside, giving me six there. So D
Dialogue: 0,0:32:40.100,0:32:42.53,Default,,0000,0000,0000,,is equal to.
Dialogue: 0,0:32:43.65,0:32:49.99,Default,,0000,0000,0000,,Dividing both sides by 8,\Nsix over 8 or 3/4 so I know
Dialogue: 0,0:32:49.99,0:32:54.87,Default,,0000,0000,0000,,everything now that I could\Npossibly want to know about
Dialogue: 0,0:32:54.87,0:32:56.34,Default,,0000,0000,0000,,this arithmetic progression.
Dialogue: 0,0:32:57.94,0:33:04.14,Default,,0000,0000,0000,,Now let's go on and have a look\Nat our second type of special
Dialogue: 0,0:33:04.14,0:33:05.47,Default,,0000,0000,0000,,sequence, a geometric
Dialogue: 0,0:33:05.47,0:33:11.40,Default,,0000,0000,0000,,progression. So.\NTake these
Dialogue: 0,0:33:11.40,0:33:14.67,Default,,0000,0000,0000,,two six
Dialogue: 0,0:33:14.67,0:33:20.68,Default,,0000,0000,0000,,1854. Let's have a look\Nat how this sequence of numbers
Dialogue: 0,0:33:20.68,0:33:23.21,Default,,0000,0000,0000,,is growing. We have two. Then we
Dialogue: 0,0:33:23.21,0:33:31.06,Default,,0000,0000,0000,,have 6. And then we have\N18. Well 326 and three sixes
Dialogue: 0,0:33:31.06,0:33:38.90,Default,,0000,0000,0000,,are 18 and three eighteens are\N54. So this sequence is growing
Dialogue: 0,0:33:38.90,0:33:45.43,Default,,0000,0000,0000,,by multiplying by three each\Ntime. What about this sequence
Dialogue: 0,0:33:45.43,0:33:48.44,Default,,0000,0000,0000,,one? Minus
Dialogue: 0,0:33:48.44,0:33:52.22,Default,,0000,0000,0000,,2 four. Minus
Dialogue: 0,0:33:52.22,0:33:57.04,Default,,0000,0000,0000,,8. What's happening here? We can\Nsee the signs are alternating,
Dialogue: 0,0:33:57.04,0:33:58.100,Default,,0000,0000,0000,,but let's just look at the
Dialogue: 0,0:33:58.100,0:34:05.07,Default,,0000,0000,0000,,numbers. 1 * 2 would be two 2 *\N2 would be four. 2 * 4 would be
Dialogue: 0,0:34:05.07,0:34:10.89,Default,,0000,0000,0000,,8. But if we made that minus\Ntwo, then one times minus two
Dialogue: 0,0:34:10.89,0:34:17.94,Default,,0000,0000,0000,,would be minus 2 - 2 times minus\Ntwo would be plus 4 + 4 times by
Dialogue: 0,0:34:17.94,0:34:23.34,Default,,0000,0000,0000,,minus two would be minus 8, so\Nthis sequence to be generated is
Dialogue: 0,0:34:23.34,0:34:27.90,Default,,0000,0000,0000,,being multiplied by minus two.\NEach term is multiplied by minus
Dialogue: 0,0:34:27.90,0:34:30.40,Default,,0000,0000,0000,,two to give the next term.
Dialogue: 0,0:34:31.21,0:34:36.84,Default,,0000,0000,0000,,These are examples of geometric\Nprogressions, or if you like,
Dialogue: 0,0:34:36.84,0:34:42.83,Default,,0000,0000,0000,,GPS. Let's try and write one\Ndown in general using some
Dialogue: 0,0:34:42.83,0:34:48.81,Default,,0000,0000,0000,,algebra. So like the AP, we take\NA to be the first term.
Dialogue: 0,0:34:49.64,0:34:54.25,Default,,0000,0000,0000,,Now we need something like D.\NThe common difference, but what
Dialogue: 0,0:34:54.25,0:35:00.12,Default,,0000,0000,0000,,we use is the letter R and we\Ncall it the common ratio, and
Dialogue: 0,0:35:00.12,0:35:05.14,Default,,0000,0000,0000,,that's the number that does the\Nmultiplying of each term to give
Dialogue: 0,0:35:05.14,0:35:06.40,Default,,0000,0000,0000,,the next term.
Dialogue: 0,0:35:07.09,0:35:14.40,Default,,0000,0000,0000,,So 3 times by two gives us 6,\Nso that's the R. In this case
Dialogue: 0,0:35:14.40,0:35:18.29,Default,,0000,0000,0000,,the three. So we do a Times by
Dialogue: 0,0:35:18.29,0:35:24.100,Default,,0000,0000,0000,,R. And then we multiply by, in\Nthis case by three again 3 times
Dialogue: 0,0:35:24.100,0:35:29.96,Default,,0000,0000,0000,,by 6 gives 18, so we multiply by\NR again, AR squared.
Dialogue: 0,0:35:30.67,0:35:38.12,Default,,0000,0000,0000,,And then we multiply by three\Nagain to give us the 54.
Dialogue: 0,0:35:38.12,0:35:41.85,Default,,0000,0000,0000,,So by our again AR cubed.
Dialogue: 0,0:35:42.68,0:35:49.64,Default,,0000,0000,0000,,And what's our end term in this\Ncase? While A is the first term
Dialogue: 0,0:35:49.64,0:35:56.60,Default,,0000,0000,0000,,8 times by R, is the second term\N8 times by R-squared is the
Dialogue: 0,0:35:56.60,0:36:03.55,Default,,0000,0000,0000,,third term 8 times by R cubed?\NIs the fourth term, so it's a
Dialogue: 0,0:36:03.55,0:36:10.02,Default,,0000,0000,0000,,times by R to the N minus one.\NBecause this power there's a
Dialogue: 0,0:36:10.02,0:36:12.100,Default,,0000,0000,0000,,one. There is always one less.
Dialogue: 0,0:36:13.00,0:36:17.27,Default,,0000,0000,0000,,And the number of the term,\Nthen its position in the
Dialogue: 0,0:36:17.27,0:36:22.70,Default,,0000,0000,0000,,sequence. And this is the end\Nterm, so it's a Times my R to
Dialogue: 0,0:36:22.70,0:36:24.25,Default,,0000,0000,0000,,the N minus one.
Dialogue: 0,0:36:25.47,0:36:31.77,Default,,0000,0000,0000,,What about adding up a\Ngeometric progression? Let's
Dialogue: 0,0:36:31.77,0:36:39.64,Default,,0000,0000,0000,,write that down. SN is\Nequal to a plus R
Dialogue: 0,0:36:39.64,0:36:41.100,Default,,0000,0000,0000,,Plus R-squared Plus.
Dialogue: 0,0:36:42.58,0:36:50.07,Default,,0000,0000,0000,,Plus AR to the N minus one,\Nand that's the sum of N terms.
Dialogue: 0,0:36:50.81,0:36:56.34,Default,,0000,0000,0000,,Going to use another trick\Nsimilar but not the same to what
Dialogue: 0,0:36:56.34,0:37:00.49,Default,,0000,0000,0000,,we did with arithmetic\Nprogressions. What I'm going to
Dialogue: 0,0:37:00.49,0:37:05.56,Default,,0000,0000,0000,,do is I'm going to multiply\Neverything by the common ratio.
Dialogue: 0,0:37:06.59,0:37:11.89,Default,,0000,0000,0000,,So I've multiplied SN by are\Ngoing to multiply this one by R,
Dialogue: 0,0:37:11.89,0:37:17.20,Default,,0000,0000,0000,,but I'm not going to write the\Nanswer there. I'm going to write
Dialogue: 0,0:37:17.20,0:37:23.73,Default,,0000,0000,0000,,it here so I've a Times by R and\NI've written it there plus now I
Dialogue: 0,0:37:23.73,0:37:29.03,Default,,0000,0000,0000,,multiply this one by R and that\Nwould give me a R-squared. I'm
Dialogue: 0,0:37:29.03,0:37:31.07,Default,,0000,0000,0000,,going to write it there.
Dialogue: 0,0:37:31.63,0:37:36.09,Default,,0000,0000,0000,,So that term is being multiplied\Nby R and it's gone to their
Dialogue: 0,0:37:36.09,0:37:40.20,Default,,0000,0000,0000,,that's being multiplied by R and\Nit's gone to their. This one
Dialogue: 0,0:37:40.20,0:37:45.35,Default,,0000,0000,0000,,will be multiplied by R and it\Nwill be a R cubed and it will
Dialogue: 0,0:37:45.35,0:37:46.72,Default,,0000,0000,0000,,have gone to their.
Dialogue: 0,0:37:47.36,0:37:52.43,Default,,0000,0000,0000,,Plus etc plus, and we think\Nabout what's happening.
Dialogue: 0,0:37:53.35,0:37:58.25,Default,,0000,0000,0000,,That term will come to here and\Nit will look just like that one.
Dialogue: 0,0:37:59.01,0:38:03.58,Default,,0000,0000,0000,,Plus and then we need to\Nmultiply this by R, and that's
Dialogue: 0,0:38:03.58,0:38:07.01,Default,,0000,0000,0000,,another. Are that we're\Nmultiplying by, so that means
Dialogue: 0,0:38:07.01,0:38:09.30,Default,,0000,0000,0000,,that becomes AR to the N.
Dialogue: 0,0:38:10.23,0:38:16.55,Default,,0000,0000,0000,,Now look at why I've lined these\Nup AR, AR, AR squared. Our
Dialogue: 0,0:38:16.55,0:38:19.46,Default,,0000,0000,0000,,squared, al, cubed, cubed and so
Dialogue: 0,0:38:19.46,0:38:25.42,Default,,0000,0000,0000,,on. So let's take these two\Nlines of algebra away from each
Dialogue: 0,0:38:25.42,0:38:31.62,Default,,0000,0000,0000,,other, so I'll have SN minus R\Ntimes by SN is equal to. Now
Dialogue: 0,0:38:31.62,0:38:38.27,Default,,0000,0000,0000,,have nothing here to take away\Nfrom a, so the a stays as it is.
Dialogue: 0,0:38:38.27,0:38:42.70,Default,,0000,0000,0000,,Then I've AR takeaway are, well,\Nthat's nothing. A R-squared
Dialogue: 0,0:38:42.70,0:38:46.24,Default,,0000,0000,0000,,takeaway R-squared? That's\Nnothing again, same there. And
Dialogue: 0,0:38:46.24,0:38:50.23,Default,,0000,0000,0000,,so on and so on. AR to the N
Dialogue: 0,0:38:50.23,0:38:55.05,Default,,0000,0000,0000,,minus one. Take away a art. The\Nend minus one nothing and then
Dialogue: 0,0:38:55.05,0:38:57.53,Default,,0000,0000,0000,,at the end I have nothing there
Dialogue: 0,0:38:57.53,0:39:02.02,Default,,0000,0000,0000,,take away.\NAR to the N.
Dialogue: 0,0:39:03.24,0:39:07.05,Default,,0000,0000,0000,,Now I need to look closely at\Nboth sides of what I've got
Dialogue: 0,0:39:07.05,0:39:10.86,Default,,0000,0000,0000,,written down, and I'm going to\Nturn this over and write it down
Dialogue: 0,0:39:10.86,0:39:18.52,Default,,0000,0000,0000,,again. So we've SN minus\NRSN is equal to A.
Dialogue: 0,0:39:19.02,0:39:22.24,Default,,0000,0000,0000,,Minus AR to the N.
Dialogue: 0,0:39:22.93,0:39:28.43,Default,,0000,0000,0000,,Now here I've got a common\Nfactor SN the some of the end
Dialogue: 0,0:39:28.43,0:39:34.35,Default,,0000,0000,0000,,terms when I take that out, I've\Nwon their minus R of them there,
Dialogue: 0,0:39:34.35,0:39:41.54,Default,,0000,0000,0000,,so I get SN times by one minus R\Nis equal 2 and here I've got a
Dialogue: 0,0:39:41.54,0:39:48.31,Default,,0000,0000,0000,,common Factor A and I can take a\Nout giving me one minus R to the
Dialogue: 0,0:39:48.31,0:39:53.81,Default,,0000,0000,0000,,N. Remember it was the sum of N\Nterms that I wanted so.
Dialogue: 0,0:39:53.86,0:40:00.47,Default,,0000,0000,0000,,SN is equal to a Times 1 minus R\Nto the N and to get the SN on
Dialogue: 0,0:40:00.47,0:40:05.60,Default,,0000,0000,0000,,its own, I've had to divide by\None minus R, so I must divide
Dialogue: 0,0:40:05.60,0:40:07.44,Default,,0000,0000,0000,,this by one minus R.
Dialogue: 0,0:40:09.37,0:40:15.84,Default,,0000,0000,0000,,And that's my formula for the\Nsum of N terms of a geometric
Dialogue: 0,0:40:15.84,0:40:19.83,Default,,0000,0000,0000,,progression. And let's just\Nremind ourselves what the
Dialogue: 0,0:40:19.83,0:40:24.81,Default,,0000,0000,0000,,symbols are N is equal to the\Nnumber of terms.
Dialogue: 0,0:40:24.82,0:40:30.85,Default,,0000,0000,0000,,A is the first\Nterm of our
Dialogue: 0,0:40:30.85,0:40:34.29,Default,,0000,0000,0000,,geometric\Nprogression and are
Dialogue: 0,0:40:34.29,0:40:40.32,Default,,0000,0000,0000,,we said was called\Nthe common ratio.
Dialogue: 0,0:40:41.42,0:40:48.01,Default,,0000,0000,0000,,OK, and let's just remember the\NNTH term in the sequence was AR
Dialogue: 0,0:40:48.01,0:40:55.11,Default,,0000,0000,0000,,to the N minus one. So those\Nare our fax so far about GPS
Dialogue: 0,0:40:55.11,0:41:00.69,Default,,0000,0000,0000,,or geometric progressions. Let's\Nsee if we can use these facts
Dialogue: 0,0:41:00.69,0:41:07.28,Default,,0000,0000,0000,,in order to be able to help\Nus solve some problems and do
Dialogue: 0,0:41:07.28,0:41:15.03,Default,,0000,0000,0000,,some questions. So first of all,\Nlet's take this 2 + 6 +
Dialogue: 0,0:41:15.03,0:41:22.27,Default,,0000,0000,0000,,18 + 54 plus. Let's say there\Nare six terms. What's the answer
Dialogue: 0,0:41:22.27,0:41:25.61,Default,,0000,0000,0000,,when it comes to adding those
Dialogue: 0,0:41:25.61,0:41:32.55,Default,,0000,0000,0000,,up? Well, we know that a\Nis equal to two. We know that
Dialogue: 0,0:41:32.55,0:41:39.82,Default,,0000,0000,0000,,our is equal to three and we\Nknow that N is equal to six. So
Dialogue: 0,0:41:39.82,0:41:47.58,Default,,0000,0000,0000,,to solve that, all we need to do\Nis write down that the sum of N
Dialogue: 0,0:41:47.58,0:41:54.86,Default,,0000,0000,0000,,terms is a Times 1 minus R to\Nthe N all over 1 minus R.
Dialogue: 0,0:41:54.86,0:41:57.28,Default,,0000,0000,0000,,Substitute our numbers in two
Dialogue: 0,0:41:57.28,0:42:03.90,Default,,0000,0000,0000,,times. 1 - 3\Nto the power 6.
Dialogue: 0,0:42:04.52,0:42:07.99,Default,,0000,0000,0000,,Over 1 -
Dialogue: 0,0:42:07.99,0:42:14.61,Default,,0000,0000,0000,,3. So this is 2 * 1 -\N3 to the power six over minus
Dialogue: 0,0:42:14.61,0:42:20.04,Default,,0000,0000,0000,,two, and we can cancel a minus\Ntwo with the two that we leave
Dialogue: 0,0:42:20.04,0:42:24.70,Default,,0000,0000,0000,,as with a minus one there and\None there if I multiply
Dialogue: 0,0:42:24.70,0:42:29.74,Default,,0000,0000,0000,,throughout by the minus one,\NI'll have minus 1 * 1 is minus
Dialogue: 0,0:42:29.74,0:42:35.95,Default,,0000,0000,0000,,one and minus one times minus 3\Nto the six is 3 to the 6th, so
Dialogue: 0,0:42:35.95,0:42:39.44,Default,,0000,0000,0000,,the sum of N terms is 3 to the
Dialogue: 0,0:42:39.44,0:42:45.43,Default,,0000,0000,0000,,power 6. Minus one and with a\NCalculator we could workout what
Dialogue: 0,0:42:45.43,0:42:49.06,Default,,0000,0000,0000,,3 to the power 6 - 1
Dialogue: 0,0:42:49.06,0:42:52.26,Default,,0000,0000,0000,,was. Let's take
Dialogue: 0,0:42:52.26,0:42:59.44,Default,,0000,0000,0000,,another. Question to do with\Nsumming the terms of a geometric
Dialogue: 0,0:42:59.44,0:43:05.89,Default,,0000,0000,0000,,progression. What's the sum\Nof that? Let's say for five
Dialogue: 0,0:43:05.89,0:43:11.77,Default,,0000,0000,0000,,terms. While we can begin by\Nidentifying the first term,
Dialogue: 0,0:43:11.77,0:43:15.89,Default,,0000,0000,0000,,that's eight, and what's the\Ncommon ratio?
Dialogue: 0,0:43:17.10,0:43:23.10,Default,,0000,0000,0000,,Well, to go from 8 to 4 as a\Nnumber we would have it, but
Dialogue: 0,0:43:23.10,0:43:27.90,Default,,0000,0000,0000,,there's a minus sign in there.\NSo that suggests that the common
Dialogue: 0,0:43:27.90,0:43:32.70,Default,,0000,0000,0000,,ratio is minus 1/2. Let's just\Ncheck it minus four times. By
Dialogue: 0,0:43:32.70,0:43:39.10,Default,,0000,0000,0000,,minus 1/2 is plus 2 + 2 times Y\Nminus 1/2 is minus one, and we
Dialogue: 0,0:43:39.10,0:43:41.90,Default,,0000,0000,0000,,said five terms, so Ann is equal
Dialogue: 0,0:43:41.90,0:43:49.49,Default,,0000,0000,0000,,to 5. So we can write\Ndown our formula. SN is equal to
Dialogue: 0,0:43:49.49,0:43:57.39,Default,,0000,0000,0000,,a Times 1 minus R to the\Npower N all over 1 minus R.
Dialogue: 0,0:43:57.98,0:44:01.12,Default,,0000,0000,0000,,And so A is 8.
Dialogue: 0,0:44:02.56,0:44:10.22,Default,,0000,0000,0000,,1 minus and this is\Nminus 1/2 to the power
Dialogue: 0,0:44:10.22,0:44:16.46,Default,,0000,0000,0000,,5. All over 1 minus minus\N1/2. You can see these
Dialogue: 0,0:44:16.46,0:44:19.33,Default,,0000,0000,0000,,questions get quite\Ncomplicated with the
Dialogue: 0,0:44:19.33,0:44:24.11,Default,,0000,0000,0000,,arithmetic, so you have to\Nbe very careful and you
Dialogue: 0,0:44:24.11,0:44:27.93,Default,,0000,0000,0000,,have to have a good\Nknowledge of fractions.
Dialogue: 0,0:44:29.32,0:44:37.11,Default,,0000,0000,0000,,This is 8 * 1. Now let's have\Na look at minus 1/2 to the power
Dialogue: 0,0:44:37.11,0:44:42.36,Default,,0000,0000,0000,,5. Well, I'm multiplying the\Nminus sign by itself five times,
Dialogue: 0,0:44:42.36,0:44:47.14,Default,,0000,0000,0000,,which would give me a negative\Nnumber, and I've got a minus
Dialogue: 0,0:44:47.14,0:44:51.91,Default,,0000,0000,0000,,sign there outside the bracket.\NThat's going to mean I've got 6
Dialogue: 0,0:44:51.91,0:44:57.09,Default,,0000,0000,0000,,minus signs together. Makes it\Nplus. So now I can look at the
Dialogue: 0,0:44:57.09,0:44:59.08,Default,,0000,0000,0000,,half to the power 5.
Dialogue: 0,0:44:59.67,0:45:02.57,Default,,0000,0000,0000,,Well, that's going to be one
Dialogue: 0,0:45:02.57,0:45:09.27,Default,,0000,0000,0000,,over. 248-1632 to\Nto the power
Dialogue: 0,0:45:09.27,0:45:12.70,Default,,0000,0000,0000,,five is 32.
Dialogue: 0,0:45:13.40,0:45:20.50,Default,,0000,0000,0000,,All over 1 minus minus 1/2.\NThat's 1 + 1/2. Let's write
Dialogue: 0,0:45:20.50,0:45:23.46,Default,,0000,0000,0000,,that as three over 2.
Dialogue: 0,0:45:24.54,0:45:27.17,Default,,0000,0000,0000,,So this is equal to.
Dialogue: 0,0:45:28.16,0:45:30.70,Default,,0000,0000,0000,,Now I've got 8.
Dialogue: 0,0:45:31.45,0:45:37.76,Default,,0000,0000,0000,,Times by one plus, one over 32,\Nand I'm dividing by three over 2
Dialogue: 0,0:45:37.76,0:45:42.72,Default,,0000,0000,0000,,to divide by a fraction. We\Ninvert the fraction that's two
Dialogue: 0,0:45:42.72,0:45:49.04,Default,,0000,0000,0000,,over 3 and we multiply by and we\Njust turn the page to finish
Dialogue: 0,0:45:49.04,0:45:50.39,Default,,0000,0000,0000,,this one off.
Dialogue: 0,0:45:51.11,0:45:58.84,Default,,0000,0000,0000,,So we have SN is\Nequal to 8 * 1
Dialogue: 0,0:45:58.84,0:46:06.57,Default,,0000,0000,0000,,+ 1 over 32 times\Nby 2/3 is equal to
Dialogue: 0,0:46:06.57,0:46:14.30,Default,,0000,0000,0000,,8 times by now one\Nand 132nd. Well, there are
Dialogue: 0,0:46:14.30,0:46:22.03,Default,,0000,0000,0000,,3230 seconds in one, so\Naltogether there I've got 33.
Dialogue: 0,0:46:22.03,0:46:25.26,Default,,0000,0000,0000,,30 seconds times
Dialogue: 0,0:46:25.26,0:46:30.93,Default,,0000,0000,0000,,by 2/3. And we\Ncan do some canceling threes
Dialogue: 0,0:46:30.93,0:46:35.58,Default,,0000,0000,0000,,into 30. Three will go 11 and\Nthrees into three. There goes
Dialogue: 0,0:46:35.58,0:46:42.54,Default,,0000,0000,0000,,one. Twos into two goes one and\Ntools into 32, goes 16 and 18
Dialogue: 0,0:46:42.54,0:46:48.86,Default,,0000,0000,0000,,two eight goes one and eight\Ninto 16 goes 2. So we 1 *
Dialogue: 0,0:46:48.86,0:46:55.19,Default,,0000,0000,0000,,11 * 1 that's just 11 over 2\Nbecause we've 2 times by one
Dialogue: 0,0:46:55.19,0:47:01.07,Default,,0000,0000,0000,,there. So we love Nova two or we\Nprefer five and a half.
Dialogue: 0,0:47:01.84,0:47:07.22,Default,,0000,0000,0000,,So that we've got the some of\Nthose five terms of that
Dialogue: 0,0:47:07.22,0:47:12.14,Default,,0000,0000,0000,,particular GP. Five and a half,\N11 over 2 or 5.5.
Dialogue: 0,0:47:12.68,0:47:19.78,Default,,0000,0000,0000,,But here's a different question.\NWhat if we've got the sequence
Dialogue: 0,0:47:19.78,0:47:27.51,Default,,0000,0000,0000,,248? 128 how many terms are\Nwe got? How many bits do we
Dialogue: 0,0:47:27.51,0:47:33.94,Default,,0000,0000,0000,,need to get from 2 up to\N128? Well, let's begin by
Dialogue: 0,0:47:33.94,0:47:36.09,Default,,0000,0000,0000,,identifying the first term
Dialogue: 0,0:47:36.09,0:47:39.61,Default,,0000,0000,0000,,that's two. This is.
Dialogue: 0,0:47:40.82,0:47:45.79,Default,,0000,0000,0000,,A geometric progression because\Nwe multiply by two to get each
Dialogue: 0,0:47:45.79,0:47:52.12,Default,,0000,0000,0000,,term. So the common ratio are is\N2 and what we don't know is
Dialogue: 0,0:47:52.12,0:47:59.17,Default,,0000,0000,0000,,what's N. So let's have a look.\NThis is the last term and we
Dialogue: 0,0:47:59.17,0:48:05.52,Default,,0000,0000,0000,,know our expression for the last\Nterm. 128 is equal to AR to the
Dialogue: 0,0:48:05.52,0:48:06.89,Default,,0000,0000,0000,,N minus one.
Dialogue: 0,0:48:07.70,0:48:14.43,Default,,0000,0000,0000,,So let's substituting some\Nof our information.
Dialogue: 0,0:48:14.45,0:48:22.31,Default,,0000,0000,0000,,A is 2 times by two\N4R to the N minus one.
Dialogue: 0,0:48:23.01,0:48:29.02,Default,,0000,0000,0000,,Well, we can divide both\Nsides by this two here,
Dialogue: 0,0:48:29.02,0:48:31.42,Default,,0000,0000,0000,,which will give us.
Dialogue: 0,0:48:32.57,0:48:38.24,Default,,0000,0000,0000,,64 is equal to two to the\NN minus one.
Dialogue: 0,0:48:38.87,0:48:45.17,Default,,0000,0000,0000,,I think about that it's 248\Nsixteen 3264 so I had to
Dialogue: 0,0:48:45.17,0:48:51.100,Default,,0000,0000,0000,,multiply 2 by itself six times\Nin order to get 64, so 2
Dialogue: 0,0:48:51.100,0:48:59.34,Default,,0000,0000,0000,,to the power 6, which is 64\Nis equal to 2 to the power
Dialogue: 0,0:48:59.34,0:49:06.70,Default,,0000,0000,0000,,N minus one, so six is equal\Nto N minus one, and so N
Dialogue: 0,0:49:06.70,0:49:09.84,Default,,0000,0000,0000,,is equal to 7, adding one.
Dialogue: 0,0:49:09.86,0:49:14.96,Default,,0000,0000,0000,,To each side. In other words,\Nthere were Seven terms in our.
Dialogue: 0,0:49:15.52,0:49:22.12,Default,,0000,0000,0000,,Geometric progression. Type\Nof question that's often given
Dialogue: 0,0:49:22.12,0:49:26.70,Default,,0000,0000,0000,,for geometric progressions is\Ngiven a geometric progression.
Dialogue: 0,0:49:26.70,0:49:32.99,Default,,0000,0000,0000,,How many terms do you need\Nto add together before you
Dialogue: 0,0:49:32.99,0:49:38.71,Default,,0000,0000,0000,,exceed a certain limit? So, for\Ninstance, here's a geometric.
Dialogue: 0,0:49:38.82,0:49:44.83,Default,,0000,0000,0000,,Progression. How many times of\Nthis geometric progression do we
Dialogue: 0,0:49:44.83,0:49:50.68,Default,,0000,0000,0000,,need to act together in order to\Nbe sure that the some of them
Dialogue: 0,0:49:50.68,0:49:52.36,Default,,0000,0000,0000,,will get over 20?
Dialogue: 0,0:49:53.04,0:49:58.93,Default,,0000,0000,0000,,Well, first of all, let's try\Nand identify this as a geometric
Dialogue: 0,0:49:58.93,0:50:04.33,Default,,0000,0000,0000,,progression. The first term is\Non and it looks like what's
Dialogue: 0,0:50:04.33,0:50:09.24,Default,,0000,0000,0000,,doing the multiplying. The\Ncommon ratio is 1.1. Let's just
Dialogue: 0,0:50:09.24,0:50:10.72,Default,,0000,0000,0000,,check that here.
Dialogue: 0,0:50:11.29,0:50:16.92,Default,,0000,0000,0000,,1.1 times by one point, one\Nwell. That's kind of like 11 *
Dialogue: 0,0:50:16.92,0:50:18.22,Default,,0000,0000,0000,,11 is 121.
Dialogue: 0,0:50:18.73,0:50:23.09,Default,,0000,0000,0000,,With two numbers after the\Ndecimal point in one point 1 *
Dialogue: 0,0:50:23.09,0:50:27.44,Default,,0000,0000,0000,,1.1 and with two numbers after\Nthe decimal point there. So yes,
Dialogue: 0,0:50:27.44,0:50:29.26,Default,,0000,0000,0000,,this is a geometric progression.
Dialogue: 0,0:50:30.05,0:50:36.72,Default,,0000,0000,0000,,So let's write down our formula\Nfor N terms sum of N terms
Dialogue: 0,0:50:36.72,0:50:42.36,Default,,0000,0000,0000,,is equal to a Times 1 minus\NR to the N.
Dialogue: 0,0:50:42.97,0:50:50.32,Default,,0000,0000,0000,,All over 1 minus R. We want to\Nknow what value of N is just
Dialogue: 0,0:50:50.32,0:50:53.26,Default,,0000,0000,0000,,going to take us over 20.
Dialogue: 0,0:50:53.88,0:51:00.09,Default,,0000,0000,0000,,So let's substituting some\Nnumbers. This is one for
Dialogue: 0,0:51:00.09,0:51:03.54,Default,,0000,0000,0000,,a 1 - 1.1 to
Dialogue: 0,0:51:03.54,0:51:10.82,Default,,0000,0000,0000,,the N. All over\N1 - 1.1 that
Dialogue: 0,0:51:10.82,0:51:15.95,Default,,0000,0000,0000,,has to be greater\Nthan 20.
Dialogue: 0,0:51:17.21,0:51:22.89,Default,,0000,0000,0000,,So one times by that isn't going\Nto affect what's in the
Dialogue: 0,0:51:22.89,0:51:29.98,Default,,0000,0000,0000,,brackets. That would be 1 - 1.1\Nto the N all over 1 - 1.1
Dialogue: 0,0:51:29.98,0:51:35.18,Default,,0000,0000,0000,,is minus nought. .1 that has to\Nbe greater than 20.
Dialogue: 0,0:51:36.51,0:51:42.26,Default,,0000,0000,0000,,Now if I use the minus sign\Nwisely. In other words, If I
Dialogue: 0,0:51:42.26,0:51:44.02,Default,,0000,0000,0000,,divide if you like.
Dialogue: 0,0:51:45.28,0:51:48.72,Default,,0000,0000,0000,,Minus note .1 into there as
Dialogue: 0,0:51:48.72,0:51:52.11,Default,,0000,0000,0000,,a. Division, then I'll have.
Dialogue: 0,0:51:52.69,0:51:58.88,Default,,0000,0000,0000,,The minus sign will make that a\Nminus and make that a plus, so
Dialogue: 0,0:51:58.88,0:52:05.07,Default,,0000,0000,0000,,I'll have one point 1 to the N\Nminus one and divided by North
Dialogue: 0,0:52:05.07,0:52:10.37,Default,,0000,0000,0000,,Point one is exactly the same as\Nmultiplying by 10. That means
Dialogue: 0,0:52:10.37,0:52:12.58,Default,,0000,0000,0000,,I've got a 10 here.
Dialogue: 0,0:52:13.31,0:52:16.47,Default,,0000,0000,0000,,That I can divide both sides by.
Dialogue: 0,0:52:17.62,0:52:23.61,Default,,0000,0000,0000,,So let's just write this down\Nagain 1.1 to the N minus one
Dialogue: 0,0:52:23.61,0:52:29.61,Default,,0000,0000,0000,,times by 10 has to be greater\Nthan 20. So let's divide both
Dialogue: 0,0:52:29.61,0:52:36.52,Default,,0000,0000,0000,,sides by 10, one point 1 to the\NN minus one has to be greater
Dialogue: 0,0:52:36.52,0:52:43.90,Default,,0000,0000,0000,,than two and will add the one to\Nboth sides 1.1 to the end has to
Dialogue: 0,0:52:43.90,0:52:45.74,Default,,0000,0000,0000,,be greater than three.
Dialogue: 0,0:52:46.55,0:52:52.01,Default,,0000,0000,0000,,Problem how do we find N? One of\Nthe ways of solving equations
Dialogue: 0,0:52:52.01,0:52:57.47,Default,,0000,0000,0000,,like this is to take logarithms\Nof both sides, so I'm going to
Dialogue: 0,0:52:57.47,0:53:02.51,Default,,0000,0000,0000,,take natural logarithms of both\Nsides. I'm going to do it to
Dialogue: 0,0:53:02.51,0:53:07.13,Default,,0000,0000,0000,,this site first. That's the\Nnatural logarithm of 3 N about
Dialogue: 0,0:53:07.13,0:53:12.59,Default,,0000,0000,0000,,this side. When you're taking a\Nlog of a number that's raised to
Dialogue: 0,0:53:12.59,0:53:17.21,Default,,0000,0000,0000,,the power, that's the equivalent\Nof multiplying the log of that
Dialogue: 0,0:53:17.21,0:53:23.19,Default,,0000,0000,0000,,number. By the power that's N\Ntimes the log of 1.1. Well
Dialogue: 0,0:53:23.19,0:53:29.70,Default,,0000,0000,0000,,now this is just an equation\Nfor N because N has got to
Dialogue: 0,0:53:29.70,0:53:36.22,Default,,0000,0000,0000,,be greater than the log of 3\Ndivided by the log of 1.1
Dialogue: 0,0:53:36.22,0:53:37.72,Default,,0000,0000,0000,,because after all.
Dialogue: 0,0:53:39.20,0:53:44.64,Default,,0000,0000,0000,,Log of three is just a number\Nand log of 1.1 is just a number
Dialogue: 0,0:53:44.64,0:53:48.28,Default,,0000,0000,0000,,and this is the sort of\Ncalculation that really does
Dialogue: 0,0:53:48.28,0:53:50.82,Default,,0000,0000,0000,,have to be done on a Calculator.
Dialogue: 0,0:53:51.80,0:53:57.15,Default,,0000,0000,0000,,So if we take our Calculator and\Nwe turn it on.
Dialogue: 0,0:53:58.36,0:54:02.37,Default,,0000,0000,0000,,And we do the calculation. The\Nnatural log of three.
Dialogue: 0,0:54:03.93,0:54:09.68,Default,,0000,0000,0000,,Divided by the natural log of\N1.1, we ask our Calculator to
Dialogue: 0,0:54:09.68,0:54:16.38,Default,,0000,0000,0000,,calculate that for us. It tells\Nus that it's 11.5 to 6 and some
Dialogue: 0,0:54:16.38,0:54:20.70,Default,,0000,0000,0000,,more decimal places. We're not\Nreally worried about these
Dialogue: 0,0:54:20.70,0:54:26.92,Default,,0000,0000,0000,,decimal places. An is a whole\Nnumber and it has to be greater
Dialogue: 0,0:54:26.92,0:54:33.63,Default,,0000,0000,0000,,than 11 and some bits, so N has\Ngot to be 12 or more.
Dialogue: 0,0:54:35.97,0:54:43.46,Default,,0000,0000,0000,,That's one last twist\Nto our geometric progression.
Dialogue: 0,0:54:43.46,0:54:46.90,Default,,0000,0000,0000,,Let's have a look at
Dialogue: 0,0:54:46.90,0:54:53.82,Default,,0000,0000,0000,,this one. What have\Nwe got got
Dialogue: 0,0:54:53.82,0:55:00.48,Default,,0000,0000,0000,,a geometric progression.\NFirst term a
Dialogue: 0,0:55:00.48,0:55:02.70,Default,,0000,0000,0000,,is one.
Dialogue: 0,0:55:04.36,0:55:10.91,Default,,0000,0000,0000,,Common ratio is 1/2 because\Nwe're multiplying by 1/2 each
Dialogue: 0,0:55:10.91,0:55:14.80,Default,,0000,0000,0000,,time. That write down
Dialogue: 0,0:55:14.80,0:55:21.46,Default,,0000,0000,0000,,some sums. S1, the sum\Nof the first term is just.
Dialogue: 0,0:55:22.06,0:55:25.34,Default,,0000,0000,0000,,1. What's
Dialogue: 0,0:55:25.34,0:55:32.66,Default,,0000,0000,0000,,S2? That's the\Nsum of the first 2 terms, so
Dialogue: 0,0:55:32.66,0:55:37.07,Default,,0000,0000,0000,,that's. Three over\N2.
Dialogue: 0,0:55:38.46,0:55:43.42,Default,,0000,0000,0000,,What's the sum of the first\Nthree terms? That's one.
Dialogue: 0,0:55:44.56,0:55:48.11,Default,,0000,0000,0000,,Plus 1/2 +
Dialogue: 0,0:55:48.11,0:55:55.78,Default,,0000,0000,0000,,1/4. Add those up\Nin terms of how many quarters
Dialogue: 0,0:55:55.78,0:55:59.56,Default,,0000,0000,0000,,have we got then that is
Dialogue: 0,0:55:59.56,0:56:06.70,Default,,0000,0000,0000,,7. Quarters As\Nfor the sum of
Dialogue: 0,0:56:06.70,0:56:08.50,Default,,0000,0000,0000,,the first.
Dialogue: 0,0:56:08.76,0:56:15.34,Default,,0000,0000,0000,,4. Terms.\NAdd those up in terms of how
Dialogue: 0,0:56:15.34,0:56:21.04,Default,,0000,0000,0000,,many eighths if we got so we've\Ngot eight of them there. Four of
Dialogue: 0,0:56:21.04,0:56:26.33,Default,,0000,0000,0000,,them there. That's 12. Two of\Nthem there. That's 14 and one of
Dialogue: 0,0:56:26.33,0:56:28.36,Default,,0000,0000,0000,,them there. That's 15 eighths.
Dialogue: 0,0:56:29.09,0:56:31.98,Default,,0000,0000,0000,,Seems to be some sort of\Npattern here.
Dialogue: 0,0:56:32.99,0:56:36.37,Default,,0000,0000,0000,,Here we seem to be 1/2 short of
Dialogue: 0,0:56:36.37,0:56:42.69,Default,,0000,0000,0000,,two. Here we\Nseem to
Dialogue: 0,0:56:42.69,0:56:49.41,Default,,0000,0000,0000,,be 1/4. Short of two\Nhere, we seem to be an eighth
Dialogue: 0,0:56:49.41,0:56:55.05,Default,,0000,0000,0000,,short of two and we look at the\Nfirst one. Then we're clearly 1
Dialogue: 0,0:56:55.05,0:56:56.26,Default,,0000,0000,0000,,short of two.
Dialogue: 0,0:56:56.93,0:57:04.01,Default,,0000,0000,0000,,He's a powers of two. Let's have\Na look 2 - 2 to the
Dialogue: 0,0:57:04.01,0:57:09.58,Default,,0000,0000,0000,,power zero, 'cause 2 to the\Npower zero is 1 two.
Dialogue: 0,0:57:10.31,0:57:11.51,Default,,0000,0000,0000,,Minus.
Dialogue: 0,0:57:12.75,0:57:20.66,Default,,0000,0000,0000,,2 to the power minus one 2\N- 2 to the power minus two
Dialogue: 0,0:57:20.66,0:57:24.62,Default,,0000,0000,0000,,2 - 2 to the power minus
Dialogue: 0,0:57:24.62,0:57:29.41,Default,,0000,0000,0000,,three. But each of these is\Ngetting smaller. We're getting
Dialogue: 0,0:57:29.41,0:57:34.54,Default,,0000,0000,0000,,nearer and nearer to two. The\Nnext one we take away will be a
Dialogue: 0,0:57:34.54,0:57:40.03,Default,,0000,0000,0000,,16th, the one after that will be\Na 32nd and the next bit we take
Dialogue: 0,0:57:40.03,0:57:45.88,Default,,0000,0000,0000,,off 2 is going to be a 64th and\Nthen a 128 and then at one
Dialogue: 0,0:57:45.88,0:57:50.28,Default,,0000,0000,0000,,256th. So we're getting the bits\Nwere taking away from two are
Dialogue: 0,0:57:50.28,0:57:53.57,Default,,0000,0000,0000,,getting smaller and smaller and\Nsmaller until eventually we
Dialogue: 0,0:57:53.57,0:57:56.50,Default,,0000,0000,0000,,wouldn't be able to distinguish\Nthem from zero.
Dialogue: 0,0:57:56.51,0:58:01.59,Default,,0000,0000,0000,,And so if we could Add all of\Nthese up forever, a sum to
Dialogue: 0,0:58:01.59,0:58:06.67,Default,,0000,0000,0000,,Infinity, if you like the\Nanswer, or to be 2 or as near as
Dialogue: 0,0:58:06.67,0:58:12.48,Default,,0000,0000,0000,,we want to be to two. So let's\Nsee if we can have a look at
Dialogue: 0,0:58:12.48,0:58:13.93,Default,,0000,0000,0000,,that with some algebra.
Dialogue: 0,0:58:14.76,0:58:22.21,Default,,0000,0000,0000,,We know that the sum to end\Nterms is equal to a Times 1
Dialogue: 0,0:58:22.21,0:58:27.53,Default,,0000,0000,0000,,minus R to the N all over\N1 minus R.
Dialogue: 0,0:58:28.05,0:58:33.12,Default,,0000,0000,0000,,What we want to have a look at\Nis this thing are because what
Dialogue: 0,0:58:33.12,0:58:34.57,Default,,0000,0000,0000,,was crucial about this?
Dialogue: 0,0:58:35.96,0:58:42.91,Default,,0000,0000,0000,,Geometric progression was at the\Ncommon ratio was a half a
Dialogue: 0,0:58:42.91,0:58:45.44,Default,,0000,0000,0000,,number less than one.
Dialogue: 0,0:58:45.45,0:58:48.19,Default,,0000,0000,0000,,So let's have a look what
Dialogue: 0,0:58:48.19,0:58:55.93,Default,,0000,0000,0000,,happens. When all is bigger than\None to R to the power N.
Dialogue: 0,0:58:56.63,0:59:01.78,Default,,0000,0000,0000,,We are is bigger than one and we\Nkeep multiplying it by itself.
Dialogue: 0,0:59:02.28,0:59:08.74,Default,,0000,0000,0000,,Grows, it grows very rapidly and\Nreally gets very big very
Dialogue: 0,0:59:08.74,0:59:15.78,Default,,0000,0000,0000,,quickly. Check it with two, 2,\Nfour, 816. It goes off til
Dialogue: 0,0:59:15.78,0:59:20.88,Default,,0000,0000,0000,,Infinity. And because it goes\Noff to Infinity, it takes the
Dialogue: 0,0:59:20.88,0:59:22.88,Default,,0000,0000,0000,,sum with it as well.
Dialogue: 0,0:59:24.35,0:59:28.23,Default,,0000,0000,0000,,What about if our is equal
Dialogue: 0,0:59:28.23,0:59:34.38,Default,,0000,0000,0000,,to 1? Well, we can't really\Nuse this formula then because we
Dialogue: 0,0:59:34.38,0:59:39.02,Default,,0000,0000,0000,,would be dividing by zero. But\Nif you think about it, are
Dialogue: 0,0:59:39.02,0:59:44.44,Default,,0000,0000,0000,,equals 1 means every term is the\Nsame. So if we start off with
Dialogue: 0,0:59:44.44,0:59:49.86,Default,,0000,0000,0000,,one every term is the same 1111\Nand you just add them all up.
Dialogue: 0,0:59:49.86,0:59:55.28,Default,,0000,0000,0000,,But again that means the sum is\Ngoing to go off to Infinity if
Dialogue: 0,0:59:55.28,0:59:59.54,Default,,0000,0000,0000,,you take the number any number\Nand add it to itself.
Dialogue: 0,1:00:00.34,1:00:04.50,Default,,0000,0000,0000,,An infinite number of times\Nyou're going to get a very, very
Dialogue: 0,1:00:04.50,1:00:12.08,Default,,0000,0000,0000,,big number. What happens if\Nour is less than minus one?
Dialogue: 0,1:00:12.08,1:00:14.71,Default,,0000,0000,0000,,Something like minus 2?
Dialogue: 0,1:00:15.45,1:00:19.75,Default,,0000,0000,0000,,Well, what's going to happen\Nthen to R to the N?
Dialogue: 0,1:00:20.92,1:00:26.06,Default,,0000,0000,0000,,Well, it's going to be plus an.\NIt's going to be minus as we
Dialogue: 0,1:00:26.06,1:00:31.20,Default,,0000,0000,0000,,multiply by this number such as\Nminus two. So we have minus 2 +
Dialogue: 0,1:00:31.20,1:00:35.60,Default,,0000,0000,0000,,4 minus A. The thing to notice\Nis it's getting bigger, it's
Dialogue: 0,1:00:35.60,1:00:40.37,Default,,0000,0000,0000,,getting bigger each time. So\Nagain are to the end is going to
Dialogue: 0,1:00:40.37,1:00:44.04,Default,,0000,0000,0000,,go off to Infinity. It's going\Nto oscillate between plus
Dialogue: 0,1:00:44.04,1:00:48.44,Default,,0000,0000,0000,,Infinity and minus Infinity, but\Nit's going to get very big and
Dialogue: 0,1:00:48.44,1:00:51.75,Default,,0000,0000,0000,,that means this sum is also\Ngoing to get.
Dialogue: 0,1:00:51.75,1:00:52.49,Default,,0000,0000,0000,,Very big.
Dialogue: 0,1:00:53.57,1:01:00.09,Default,,0000,0000,0000,,What about our equals minus one?\NWell, if R equals minus one,
Dialogue: 0,1:01:00.09,1:01:05.52,Default,,0000,0000,0000,,let's think about a sequence\Nlike that. Well, a typical
Dialogue: 0,1:01:05.52,1:01:08.77,Default,,0000,0000,0000,,sequence might be 1 - 1.
Dialogue: 0,1:01:09.49,1:01:14.09,Default,,0000,0000,0000,,1 - 1 and we can see the\Nproblem. It depends where we
Dialogue: 0,1:01:14.09,1:01:19.40,Default,,0000,0000,0000,,stop. If I stop here the sum is\N0 but if I put another one
Dialogue: 0,1:01:19.40,1:01:24.00,Default,,0000,0000,0000,,there, the sum is one. So we've\Ngot an infinite number of terms
Dialogue: 0,1:01:24.00,1:01:28.61,Default,,0000,0000,0000,,then. Well, it depends on money\NI've got us to what the answer
Dialogue: 0,1:01:28.61,1:01:33.56,Default,,0000,0000,0000,,is so there isn't a limit for\NSN. There isn't a thing that it
Dialogue: 0,1:01:33.56,1:01:35.69,Default,,0000,0000,0000,,can come to a definite number.
Dialogue: 0,1:01:36.67,1:01:40.96,Default,,0000,0000,0000,,Let's have a look. We've\Nconsidered all possible values
Dialogue: 0,1:01:40.96,1:01:46.69,Default,,0000,0000,0000,,of our except those where are is\Nbetween plus and minus one.
Dialogue: 0,1:01:47.23,1:01:50.27,Default,,0000,0000,0000,,Let's take our equals 1/2\Nas an example.
Dialogue: 0,1:01:51.59,1:01:54.45,Default,,0000,0000,0000,,Or half trans by half is 1/4.
Dialogue: 0,1:01:55.47,1:01:59.21,Default,,0000,0000,0000,,Reply by 1/2. Again\Nthat's an eighth.
Dialogue: 0,1:02:00.41,1:02:03.17,Default,,0000,0000,0000,,Multiply by 1/2 again, that's a
Dialogue: 0,1:02:03.17,1:02:08.85,Default,,0000,0000,0000,,16. Multiplied by 1/2 again,\Nthat's a 32nd.
Dialogue: 0,1:02:09.64,1:02:14.81,Default,,0000,0000,0000,,By half again that's a 64th by\N1/2 again, that's 128.
Dialogue: 0,1:02:16.08,1:02:21.82,Default,,0000,0000,0000,,It's getting smaller, and if we\Ndo it enough times then it's
Dialogue: 0,1:02:21.82,1:02:24.68,Default,,0000,0000,0000,,going to head off till 0.
Dialogue: 0,1:02:25.69,1:02:30.53,Default,,0000,0000,0000,,What about a negative one? You\Nmight say, let's think about
Dialogue: 0,1:02:30.53,1:02:36.18,Default,,0000,0000,0000,,minus 1/2. Now multiplied by\Nminus 1/2, it's a quarter.
Dialogue: 0,1:02:36.18,1:02:41.50,Default,,0000,0000,0000,,Multiply the quarter by minus\N1/2. It's minus an eighth.
Dialogue: 0,1:02:41.50,1:02:46.82,Default,,0000,0000,0000,,Multiply again by minus 1/2.\NWell, that's plus a 16th.
Dialogue: 0,1:02:46.82,1:02:52.67,Default,,0000,0000,0000,,Multiply again by minus 1/2.\NThat's minus a 32nd, so we're
Dialogue: 0,1:02:52.67,1:02:59.05,Default,,0000,0000,0000,,approaching 0, but where dotting\Nabout either side of 0 plus them
Dialogue: 0,1:02:59.05,1:03:03.84,Default,,0000,0000,0000,,were minus, then were plus then\Nwhere mine is.
Dialogue: 0,1:03:03.87,1:03:08.56,Default,,0000,0000,0000,,We're getting nearer to zero\Neach time, so again are to the
Dialogue: 0,1:03:08.56,1:03:13.64,Default,,0000,0000,0000,,power. N is going off to zero.\NWhat does that mean? It means
Dialogue: 0,1:03:13.64,1:03:19.12,Default,,0000,0000,0000,,that this some. Here we can have\Nwhat we call a sum to Infinity.
Dialogue: 0,1:03:19.12,1:03:23.81,Default,,0000,0000,0000,,Sometimes it's just written with\Nan S and sometimes it's got a
Dialogue: 0,1:03:23.81,1:03:25.77,Default,,0000,0000,0000,,little Infinity sign on it.
Dialogue: 0,1:03:26.48,1:03:32.82,Default,,0000,0000,0000,,What that tells us? Because this\Nart of the end is going off to
Dialogue: 0,1:03:32.82,1:03:40.07,Default,,0000,0000,0000,,0 then it's a times by one over\N1 minus R and that's our sum to
Dialogue: 0,1:03:40.07,1:03:42.79,Default,,0000,0000,0000,,Infinity. In other words, we can
Dialogue: 0,1:03:42.79,1:03:48.93,Default,,0000,0000,0000,,add up. An infinite number of\Nterms for a geometric
Dialogue: 0,1:03:48.93,1:03:55.77,Default,,0000,0000,0000,,progression provided. The common\Nratio is between one and minus
Dialogue: 0,1:03:55.77,1:04:02.80,Default,,0000,0000,0000,,one, so let's have a look\Nat an example. Supposing we've
Dialogue: 0,1:04:02.80,1:04:04.72,Default,,0000,0000,0000,,got this row.
Dialogue: 0,1:04:04.75,1:04:12.12,Default,,0000,0000,0000,,Metric progression.\NWell, first term is one
Dialogue: 0,1:04:12.12,1:04:16.52,Default,,0000,0000,0000,,now a common ratio is\N1/3.
Dialogue: 0,1:04:17.71,1:04:19.80,Default,,0000,0000,0000,,And what does this come to when
Dialogue: 0,1:04:19.80,1:04:26.46,Default,,0000,0000,0000,,we add up? As many terms as\Nwe can, what's the sum to
Dialogue: 0,1:04:26.46,1:04:32.65,Default,,0000,0000,0000,,Infinity? We know the formula\Nthat's a over 1 minus R, so
Dialogue: 0,1:04:32.65,1:04:39.36,Default,,0000,0000,0000,,let's put the numbers in this\None for a over 1 - 1/3.
Dialogue: 0,1:04:40.46,1:04:46.32,Default,,0000,0000,0000,,So the one on tops OK and the\None minus third. Well that's
Dialogue: 0,1:04:46.32,1:04:52.19,Default,,0000,0000,0000,,2/3, and if we're dividing by a\Nfraction then we invert it and
Dialogue: 0,1:04:52.19,1:04:57.15,Default,,0000,0000,0000,,multiply. So altogether that\Nwould come to three over 2, so
Dialogue: 0,1:04:57.15,1:04:59.85,Default,,0000,0000,0000,,it's very easy formula to use.
Dialogue: 0,1:05:01.51,1:05:04.25,Default,,0000,0000,0000,,Finally, just let's recap for a
Dialogue: 0,1:05:04.25,1:05:06.71,Default,,0000,0000,0000,,geometric progression. A.
Dialogue: 0,1:05:07.36,1:05:10.76,Default,,0000,0000,0000,,Is the first term.
Dialogue: 0,1:05:10.76,1:05:13.48,Default,,0000,0000,0000,,Aw.
Dialogue: 0,1:05:14.74,1:05:16.31,Default,,0000,0000,0000,,Is the common.
Dialogue: 0,1:05:17.14,1:05:24.03,Default,,0000,0000,0000,,Ratio. So a\Ngeometric progression looks like
Dialogue: 0,1:05:24.03,1:05:31.44,Default,,0000,0000,0000,,AARA, R-squared, AR, cubed and\Nthe N Terminus series AR
Dialogue: 0,1:05:31.44,1:05:35.14,Default,,0000,0000,0000,,to the N minus one.
Dialogue: 0,1:05:35.78,1:05:41.06,Default,,0000,0000,0000,,And if we want to add up this\Nsequence of numbers SN.
Dialogue: 0,1:05:41.82,1:05:49.63,Default,,0000,0000,0000,,Then that's a Times 1 minus R to\Nthe power N or over 1 minus R.
Dialogue: 0,1:05:50.49,1:05:55.35,Default,,0000,0000,0000,,And if we're lucky enough to\Nhave our between plus and
Dialogue: 0,1:05:55.35,1:05:59.77,Default,,0000,0000,0000,,minus one, sometimes that's\Nwritten as the modulus of art
Dialogue: 0,1:05:59.77,1:06:04.63,Default,,0000,0000,0000,,is less than one. If we're\Nlucky to have this condition,
Dialogue: 0,1:06:04.63,1:06:10.38,Default,,0000,0000,0000,,then we can get a sum to\NInfinity, which is a over 1
Dialogue: 0,1:06:10.38,1:06:11.26,Default,,0000,0000,0000,,minus R.