0:00:00.500,0:00:06.560
In this video, we're going to be[br]looking at sequences and series,

0:00:06.560,0:00:11.105
so let's begin by looking at[br]what a sequences.

0:00:12.030,0:00:15.885
This, for instance is a

0:00:15.885,0:00:23.280
sequence. It's a[br]set of numbers.

0:00:23.830,0:00:30.745
And here we seem to have a rule.[br]All of these are odd numbers, or

0:00:30.745,0:00:36.277
we can look at it. We increase[br]by two each time 13579.

0:00:37.280,0:00:42.460
So there's our sequence of[br]odd numbers.

0:00:43.900,0:00:45.788
Here is another sequence.

0:00:46.650,0:00:53.742
These numbers[br]are the

0:00:53.742,0:01:01.102
square numbers.[br]1 squared, 2 squared, is 4 three

0:01:01.102,0:01:07.066
squared is 9. Four squared is 16[br]and 5 squared is 25. So again

0:01:07.066,0:01:12.604
we've got a sequence of numbers.[br]We've got a rule that seems to

0:01:12.604,0:01:15.269
produce them. Those are the.

0:01:15.890,0:01:22.140
Square numbers. Is a[br]slightly different sequence.

0:01:22.510,0:01:27.450
Here we've got alternation[br]between one and minus one back

0:01:27.450,0:01:34.366
to one on again to minus one[br]back to one on again to minus

0:01:34.366,0:01:38.812
one. But this is still a[br]sequence of numbers.

0:01:39.510,0:01:42.966
Now, because I've written some[br]dots after it here.

0:01:43.660,0:01:48.769
This means that this is meant to[br]be an infinite sequence. It goes

0:01:48.769,0:01:53.485
on forever and this is meant to[br]be an infinite sequence. It

0:01:53.485,0:01:58.594
carries on forever, and this one[br]does too. If I want a finite

0:01:58.594,0:02:05.570
sequence. What might[br]a finite sequence

0:02:05.570,0:02:09.188
look like, for

0:02:09.188,0:02:15.904
instance 1359? That would[br]be a finite sequence. We've got

0:02:15.904,0:02:18.652
4 numbers and then it stops

0:02:18.652,0:02:25.160
dead. Perhaps if we look[br]at the sequence of square

0:02:25.160,0:02:30.960
numbers 1, four, 916, that again[br]is a finite sequence.

0:02:32.070,0:02:39.950
Sequence that will be very[br]interested in is the sequence

0:02:39.950,0:02:47.042
of whole numbers, the counting[br]numbers, the integers. So

0:02:47.042,0:02:54.922
there's a sequence of integers,[br]and it's finite because it

0:02:54.922,0:03:02.802
stops at N, so we're[br]counting 123456789 up to N.

0:03:02.820,0:03:08.150
And the length of this sequence[br]is an the integer, the number N.

0:03:10.310,0:03:16.277
Very popular sequence of[br]numbers. Quite well known is

0:03:16.277,0:03:18.266
this particular sequence.

0:03:19.000,0:03:23.610
This is a slightly different[br]sequence. It's infinite keeps on

0:03:23.610,0:03:25.915
going and it's called the

0:03:25.915,0:03:30.714
Fibonacci sequence. And we can[br]see how it's generated. This

0:03:30.714,0:03:35.823
number 2 is formed by adding the[br]one and the one together, and

0:03:35.823,0:03:40.539
then the three is formed by[br]adding the one and the two

0:03:40.539,0:03:45.255
together. The Five is formed by[br]adding the two and the three

0:03:45.255,0:03:46.827
together, and so on.

0:03:47.840,0:03:52.585
So there's a question here. How[br]could we write this rule down in

0:03:52.585,0:03:56.600
general, where we can say it[br]that any particular term is

0:03:56.600,0:04:00.615
generated by adding the two[br]numbers that come before it in

0:04:00.615,0:04:04.630
the sequence together? But how[br]might we set that down? How

0:04:04.630,0:04:09.740
might we label it? One way might[br]be to use algebra and say will

0:04:09.740,0:04:14.120
call the first term you, and[br]because it's the first time we

0:04:14.120,0:04:17.040
want to label it, so we call it

0:04:17.040,0:04:20.690
you one. And then the next[br]terminal sequence, the second

0:04:20.690,0:04:25.305
term. It would make sense to[br]call it you too, and the third

0:04:25.305,0:04:29.565
term in our sequence. It would[br]make sense therefore, to call it

0:04:29.565,0:04:36.916
you 3. You four and so on[br]up to UN. So this represents a

0:04:36.916,0:04:43.012
finite sequence that's got N[br]terms in it. If we look at

0:04:43.012,0:04:48.600
the Fibonacci sequence as an[br]example of making use of this

0:04:48.600,0:04:54.696
kind of notation, we could say[br]that the end term UN was

0:04:54.696,0:04:59.776
generated by adding together the[br]two terms that come immediately

0:04:59.776,0:05:01.300
before it will.

0:05:01.320,0:05:05.110
Term that comes immediately[br]before this must have a number

0:05:05.110,0:05:10.416
attached to it. That's one less[br]than N and that would be N minus

0:05:10.416,0:05:15.910
one. Plus on the term that's[br]down, the term that comes

0:05:15.910,0:05:20.926
immediately before this one must[br]have a number attached to it.

0:05:20.926,0:05:26.854
That's one less than that. Well,[br]that's UN minus 1 - 1 taking

0:05:26.854,0:05:30.046
away 2 ones were taking away two

0:05:30.046,0:05:36.642
altogether. So that we can see[br]how we might use the algebra

0:05:36.642,0:05:42.546
this algebraic notation help us[br]write down a rule for the Fibo

0:05:42.546,0:05:46.406
Nachi sequence. OK, how can

0:05:46.406,0:05:52.185
we? Use this in a[br]slightly different way.

0:05:52.710,0:06:00.270
What we need to look at now[br]is to move on and have a

0:06:00.270,0:06:04.050
look what we mean by a series.

0:06:04.630,0:06:11.716
This is a[br]sequence, label it.

0:06:12.490,0:06:17.330
A sequence it's a list of[br]numbers generated by some

0:06:17.330,0:06:22.170
particular rule. It's finite[br]because there are any of them.

0:06:22.900,0:06:28.510
What then, is a series series is[br]what we get.

0:06:29.280,0:06:32.970
When we add.

0:06:32.970,0:06:36.478
Terms of the sequence.

0:06:37.070,0:06:44.078
Together And because[br]we're adding together and terms

0:06:44.078,0:06:46.934
will call this SN.

0:06:47.860,0:06:55.610
The sum of N terms[br]and it's that which is

0:06:55.610,0:06:57.160
the series.

0:06:59.040,0:07:06.097
So. Let's have a[br]look at the sequence

0:07:06.097,0:07:12.825
of numbers 123456, and[br]so on up to

0:07:12.825,0:07:13.666
N.

0:07:15.080,0:07:17.190
Then S1.

0:07:18.850,0:07:22.210
Is just one.

0:07:22.840,0:07:24.700
S2.

0:07:25.750,0:07:32.660
Is the sum of the[br]first 2 terms 1 +

0:07:32.660,0:07:34.980
2? And that gives us 3.

0:07:35.740,0:07:42.893
S3. Is[br]the sum of the first three

0:07:42.893,0:07:46.607
terms 1 + 2 + 3?

0:07:47.220,0:07:50.805
And that gives us 6

0:07:50.805,0:07:58.250
and S4. Is the sum[br]of the first four terms 1 +

0:07:58.250,0:08:02.100
2 + 3 + 4 and that

0:08:02.100,0:08:03.580
gives us. Hey.

0:08:04.480,0:08:10.148
So this gives us the basic[br]vocabulary to be able to move on

0:08:10.148,0:08:14.944
to the next section of the[br]video, but just let's remind

0:08:14.944,0:08:16.688
ourselves first of all.

0:08:17.220,0:08:18.640
A sequence.

0:08:20.180,0:08:24.203
Is a set of numbers[br]generated by some rule.

0:08:25.830,0:08:31.402
A series is what we get when we[br]add the terms of the sequence

0:08:31.402,0:08:37.516
together. This particular[br]sequence has N terms in it

0:08:37.516,0:08:43.486
because we've labeled each term[br]in the sequence with accounting

0:08:43.486,0:08:49.456
number. If you like U1U2U free,[br]you fall you N.

0:08:51.380,0:08:55.921
Now. With this vocabulary of[br]sequences and series in mind,

0:08:55.921,0:09:01.756
we're going to go on and have a[br]look at a 2 special kinds of

0:09:01.756,0:09:06.630
sequences. The first one is[br]called an arithmetic progression

0:09:06.630,0:09:09.990
and the second one is called a

0:09:09.990,0:09:15.125
geometric progression. Will[br]begin with an arithmetic

0:09:15.125,0:09:22.062
progression. Let's start by[br]having a look at this

0:09:22.062,0:09:24.950
sequence of. Odd

0:09:25.470,0:09:32.748
Numbers that we[br]had before 1357.

0:09:33.300,0:09:40.026
Is another sequence[br]not 1020 thirty,

0:09:40.026,0:09:43.389
and so on.

0:09:44.640,0:09:51.634
What we can see in this first[br]sequence is that each term after

0:09:51.634,0:09:56.476
the first one is formed by[br]adding on to.

0:09:57.210,0:10:00.258
1 + 2 gives us 3.

0:10:00.840,0:10:03.858
3 + 2 gives us 5.

0:10:04.710,0:10:11.145
5 + 2 gives us 7 and it's[br]because we're adding on the

0:10:11.145,0:10:17.085
same amount every time. This[br]is an example of what we call

0:10:17.085,0:10:18.075
an arithmetic.

0:10:19.870,0:10:28.082
Progression.[br]If we look at this sequence of

0:10:28.082,0:10:33.538
numbers, we can see exactly the[br]same property we've started with

0:10:33.538,0:10:39.986
zero. We've added on 10, and[br]we've added on 10 again to get

0:10:39.986,0:10:46.930
20. We've had it on 10 again to[br]get 30, so again, this is

0:10:46.930,0:10:49.410
exactly the same. It's an

0:10:49.410,0:10:56.166
arithmetic progression. We don't[br]have to add on things, so

0:10:56.166,0:11:03.283
for instance a sequence of[br]numbers that went like this 8

0:11:03.283,0:11:05.871
five, 2 - 1.

0:11:06.560,0:11:08.665
Minus

0:11:08.665,0:11:14.253
4. If we look what's[br]happening where going from 8:00

0:11:14.253,0:11:18.510
to 5:00, so that's takeaway[br]three were going from five to

0:11:18.510,0:11:23.154
two, so that's takeaway. Three[br]were going from 2 to minus. One

0:11:23.154,0:11:27.411
takeaway. Three were going from[br]minus one to minus four takeaway

0:11:27.411,0:11:33.662
3. Another way of thinking about[br]takeaway three is to say where

0:11:33.662,0:11:35.510
adding on minus three.

0:11:36.330,0:11:43.050
8 at minus three is 5 five at[br]minus three is 2, two AD minus

0:11:43.050,0:11:48.426
three is minus one, so again,[br]this is an example of an

0:11:48.426,0:11:54.084
arithmetic progression. And what[br]we want to be able to do is to

0:11:54.084,0:11:56.996
try and encapsulate this[br]arithmetic progression in some

0:11:56.996,0:11:59.180
algebra, so we'll use the letter

0:11:59.180,0:12:04.686
A. To stand for[br]the first term.

0:12:05.310,0:12:12.360
And will use the letter[br]D to stand for the

0:12:12.360,0:12:17.768
common difference. Now the[br]common difference is the

0:12:17.768,0:12:23.686
difference between each term and[br]it's called common because it is

0:12:23.686,0:12:30.142
common to each between each[br]term. So let's have a look at

0:12:30.142,0:12:36.598
one 357 and let's have a think[br]about how it's structured 13.

0:12:37.340,0:12:44.820
5. 7 and so[br]on. So we begin with one and

0:12:44.820,0:12:48.817
then the three is 1 + 2.

0:12:50.270,0:12:57.755
The Five is 1 + 2 tools because[br]by the time we got to five,

0:12:57.755,0:13:00.749
we've added four onto the one.

0:13:01.420,0:13:07.894
The Seven is one plus. Now the[br]time we've got to Seven, we've

0:13:07.894,0:13:14.368
added three tools on. Let's just[br]do one more. Let's put nine in

0:13:14.368,0:13:19.846
there and that would be 1 + 4[br]times by two.

0:13:20.980,0:13:25.320
So let's see if we can begin to[br]write this down. This is one.

0:13:26.280,0:13:33.235
Now what have we got here? This[br]is the second term in the

0:13:33.235,0:13:39.773
series. But we've only got 1 two[br]there, so if you like we've got

0:13:39.773,0:13:43.141
1 + 2 - 1 times by two.

0:13:44.060,0:13:48.776
One plus now, what's multiplying[br]the two here? Well, this is the

0:13:48.776,0:13:50.741
third term in the series.

0:13:51.650,0:13:59.306
So we've got a 2 here,[br]so we're multiplying by 3 -

0:13:59.306,0:14:05.199
1. Here this is term[br]#4 and we're

0:14:05.199,0:14:10.932
multiplying by three,[br]so that's 4 - 1 times

0:14:10.932,0:14:17.302
by two. And here this[br]is term #5, so we've

0:14:17.302,0:14:23.035
got 1 + 5 - 1[br]times by two.

0:14:24.430,0:14:28.670
Now, if we think about[br]what's happening here.

0:14:31.120,0:14:33.268
We're starting with A.

0:14:34.670,0:14:37.610
And then on to the A. We're

0:14:37.610,0:14:44.940
adding D. Then we're adding on[br]another day, so that's a plus

0:14:44.940,0:14:51.828
2D, and then we're adding on[br]another D. So that's a plus

0:14:51.828,0:14:57.214
3D. The question is, if we've[br]got N terms in our sequence,

0:14:57.214,0:15:02.450
then what's the last term? But[br]if we look, we can see that the

0:15:02.450,0:15:04.320
first term was just a.

0:15:04.880,0:15:12.244
The second term was a plus, one[br]D. The third term was a plus

0:15:12.244,0:15:19.608
2D. The fourth term was a plus[br]3D, so the end term must be

0:15:19.608,0:15:22.238
a plus N minus one.

0:15:22.750,0:15:23.230
Gay.

0:15:25.200,0:15:32.150
Now, this last term of[br]our sequence, we often label

0:15:32.150,0:15:35.625
L and call it the

0:15:35.625,0:15:37.015
last term.

0:15:37.070,0:15:39.650
Or

0:15:40.420,0:15:42.530
The end.

0:15:43.140,0:15:46.750
Turn. To be more mathematical

0:15:46.750,0:15:51.368
about it. And one of the things[br]that we'd like to be able to do

0:15:51.368,0:15:54.566
with a sequence of numbers like[br]this is get to a series. In

0:15:54.566,0:15:58.256
other words, to be able to add[br]them up. So let's have a look at

0:15:58.256,0:16:05.693
that. So SN the some of these[br]end terms is A plus A+B plus

0:16:05.693,0:16:12.847
A plus 2B plus. But I want[br]just to stop there and what I

0:16:12.847,0:16:19.490
want to do is I want to[br]start at the end. This end

0:16:19.490,0:16:24.089
now now the last one will be[br]plus L.

0:16:25.150,0:16:29.518
So what will be the next one[br]back when we generate each term

0:16:29.518,0:16:35.230
by adding on D. So we added on D[br]to this one to get L. So this

0:16:35.230,0:16:37.582
one's got to be L minus D.

0:16:38.920,0:16:45.750
And the one before that[br]one similarly will be L

0:16:45.750,0:16:51.520
minus 2D. On the rest of the[br]terms will be in between.

0:16:52.640,0:16:54.999
Now I'm going to use a trick.

0:16:55.540,0:16:58.850
Mathematicians often use. I'm[br]going to write this down the

0:16:58.850,0:17:02.072
other way around. So I have L

0:17:02.072,0:17:05.870
there. Plus L minus

0:17:05.870,0:17:12.433
D. Plus L minus 2D plus[br]plus. Now what will I have?

0:17:12.433,0:17:17.869
Well, writing this down either[br]way around, I'll Have A at the

0:17:17.869,0:17:21.255
end. Then I'll have this next

0:17:21.255,0:17:23.730
term a. Plus D.

0:17:24.230,0:17:28.973
And I'll have this next[br]term, A plus 2D.

0:17:31.490,0:17:36.319
Now I'm going to add these two[br]together. Let's look what

0:17:36.319,0:17:41.587
happens if I add SN&SN together.[br]I've just got two of them.

0:17:42.970,0:17:49.548
By ad A&L together I get a[br]plus L let me just group

0:17:49.548,0:17:50.560
those together.

0:17:51.820,0:17:59.004
Now I've got a plus D&L Minus D,[br]so if I add them together I have

0:17:59.004,0:18:06.188
a plus L Plus D minus D, so[br]all I've got left is A plus L.

0:18:07.290,0:18:14.192
But the same thing is going to[br]happen here. I have a plus L

0:18:14.192,0:18:19.122
Plus 2D Takeaway 2D, so again[br]just a plus L.

0:18:20.030,0:18:25.070
When we get down To this end,[br]it's still the same thing

0:18:25.070,0:18:30.110
happening. I've A plus L[br]takeaway 2D add onto D so again

0:18:30.110,0:18:35.570
the DS have disappeared. If you[br]like and I've got L plus A.

0:18:36.480,0:18:43.884
Plus a plusle takeaway D add[br]on DLA and right at the

0:18:43.884,0:18:46.969
end. L plus a again.

0:18:48.640,0:18:53.333
Well, how many of these have I[br]got? But I've got N terms.

0:18:54.130,0:19:01.732
In each of these lines of sums,[br]so I must still have end terms

0:19:01.732,0:19:07.705
here, and so this must be an[br]times a plus L.

0:19:08.390,0:19:15.530
And so if we now divide[br]both sides by two, we have.

0:19:15.530,0:19:22.670
SN is 1/2 of N times[br]by a plus L and that

0:19:22.670,0:19:28.620
gives us our some of the[br]terms of an arithmetic

0:19:28.620,0:19:35.409
progression. Let's just write[br]down again the two results that

0:19:35.409,0:19:42.741
we've got. We've got L the[br]end term, or the final term

0:19:42.741,0:19:50.073
is equal to a plus N[br]minus one times by D and

0:19:50.073,0:19:53.739
we've got the SN is 1/2.

0:19:54.320,0:20:01.380
Times by N number of[br]terms times by a plus

0:20:01.380,0:20:02.086
L.

0:20:03.290,0:20:09.062
Now, one thing we can do is take[br]this expression for L and

0:20:09.062,0:20:10.838
substitute it into here.

0:20:11.770,0:20:19.570
Replacing this al, so let's do[br]that. SN is equal to 1/2.

0:20:20.180,0:20:27.716
Times by N number of terms[br]times by a plus and instead

0:20:27.716,0:20:35.252
of L will write this a[br]plus N minus one times by

0:20:35.252,0:20:39.558
D. April say gives us[br]two way.

0:20:40.600,0:20:47.640
So the sum of the[br]end terms is 1/2 an

0:20:47.640,0:20:51.160
2A plus N minus 1D.

0:20:51.930,0:20:53.958
Close the bracket.

0:20:55.730,0:21:02.064
And these. That I'm[br]underlining are the three

0:21:02.064,0:21:05.838
important things about an[br]arithmetic progression.

0:21:07.280,0:21:11.070
If A is the first

0:21:11.070,0:21:16.826
term. And D is[br]the common difference.

0:21:17.750,0:21:23.102
And N[br]is the

0:21:23.102,0:21:27.116
number of[br]terms.

0:21:28.370,0:21:33.545
In our arithmetic progression,[br]then, this expression gives us

0:21:33.545,0:21:39.870
the NTH or the last term.[br]This expression gives us the

0:21:39.870,0:21:46.195
some of those N terms, and[br]this expression gives us also

0:21:46.195,0:21:49.645
the sum of the end terms.

0:21:50.560,0:21:54.520
One of the things that you also[br]need to understand is that

0:21:54.520,0:21:58.150
sometimes we like to shorten the[br]language as well as using

0:21:58.150,0:22:02.861
algebra. So that rather than[br]keep saying arithmetic

0:22:02.861,0:22:07.838
progression, we often refer to[br]these as a peas.

0:22:09.130,0:22:11.278
Now we've got some facts, some

0:22:11.278,0:22:16.526
information there. So let's have[br]a look at trying to see if we

0:22:16.526,0:22:18.662
can use them to solve some

0:22:18.662,0:22:25.800
questions. So let's have[br]a look at this

0:22:25.800,0:22:31.750
sequence of numbers again,[br]which we've identified.

0:22:33.460,0:22:37.016
And let's ask ourselves[br]what's the sum?

0:22:38.150,0:22:41.100
Of. The first

0:22:41.780,0:22:48.278
50 terms So[br]we could start to try and add

0:22:48.278,0:22:54.374
them up. 1 + 3 is four and four[br]and five is 9, and nine and

0:22:54.374,0:22:59.708
Seven is 16 and 16 and 9025, and[br]then the next get or getting

0:22:59.708,0:23:03.899
rather complicated. But we can[br]write down some facts about this

0:23:03.899,0:23:08.471
straight away. We can write down[br]that the first term is one.

0:23:09.070,0:23:14.350
We can write down that the[br]common difference Dean is 2 and

0:23:14.350,0:23:20.070
we can write down the number of[br]terms we're dealing with. An is

0:23:20.070,0:23:27.041
50. We know we have a[br]formula that says SN is 1/2

0:23:27.041,0:23:29.696
times the number of terms.

0:23:30.710,0:23:37.696
Times 2A plus N minus 1D. So[br]instead of having to add this up

0:23:37.696,0:23:43.684
as though it was a big[br]arithmetic sum a big problem, we

0:23:43.684,0:23:48.674
can simply substitute the[br]numbers into the formula. So SNS

0:23:48.674,0:23:52.666
50 in this case is equal to 1/2.

0:23:53.080,0:23:55.288
Times by 50.

0:23:55.800,0:24:02.940
Times by two A That's just two[br]2 * 1 plus N minus one

0:24:02.940,0:24:06.510
and is 50, so N minus one

0:24:06.510,0:24:10.210
is 49. Times by the common

0:24:10.210,0:24:11.270
difference too.

0:24:12.260,0:24:19.716
So. We[br]can cancel a 2 into the 50

0:24:19.716,0:24:23.230
that gives us 25 times by now.

0:24:23.770,0:24:30.840
2 * 49 or 2 * 49[br]is 98 and two is 100, so

0:24:30.840,0:24:37.405
we have 25 times by 100, so[br]that's 2500. So what was going

0:24:37.405,0:24:42.455
to be quite a lengthy and[br]difficult calculation's come out

0:24:42.455,0:24:48.702
quite quickly. Let's see if we[br]can solve a more difficult

0:24:48.702,0:24:53.000
problem.[br]1.

0:24:54.110,0:24:58.840
Plus 3.5.[br]+6.

0:25:00.050,0:25:03.620
Plus 8.5. Plus

0:25:04.490,0:25:08.270
Plus 101.

0:25:10.290,0:25:11.310
Add this up.

0:25:12.490,0:25:19.552
Well. Can we identify what[br]kind of a series this is? We can

0:25:19.552,0:25:25.155
see quite clearly that one to[br]3.5 while that's a gap of 2.5

0:25:25.155,0:25:32.051
and then a gap of 2.5 to 6. So[br]what we've got here is in fact

0:25:32.051,0:25:36.792
an arithmetic progression, and[br]we can see here. We've got 100

0:25:36.792,0:25:42.826
and one at the end. Our last[br]term is 101 and the first term

0:25:42.826,0:25:45.843
is one. Now we know a formula.

0:25:45.890,0:25:49.390
For the last term L.

0:25:50.050,0:25:57.133
Equals A plus N minus[br]one times by D.

0:25:58.350,0:26:05.126
Might just have a look at what[br]we know in this formula. What we

0:26:05.126,0:26:07.062
know L it's 101.

0:26:07.070,0:26:12.227
We know a It's the first[br]term, it's one.

0:26:13.280,0:26:19.259
Plus Well, we have no idea what[br]any is. We don't know how many

0:26:19.259,0:26:24.061
terms we've got, so that's N[br]minus one times by D and we know

0:26:24.061,0:26:25.776
what that is, that's 2.5.

0:26:26.510,0:26:31.790
Well, this is nothing more than[br]an equation for an, so let's

0:26:31.790,0:26:37.510
begin by taking one from each[br]side. That gives us 100 equals N

0:26:37.510,0:26:43.230
minus one times by 2.5. And now[br]I'm going to divide both sides

0:26:43.230,0:26:49.830
by 2.5 and that will give me 40[br]equals N minus one, and now I'll

0:26:49.830,0:26:57.310
add 1 to both sides and so 41 is[br]equal to end, so I know how many

0:26:57.310,0:27:04.198
terms that. Are in this series,[br]So what I can do now is I

0:27:04.198,0:27:10.186
can add it up because the sum of[br]N terms is 1/2.

0:27:10.860,0:27:13.872
NA plus

0:27:13.872,0:27:20.800
L. And I[br]now know all these terms

0:27:20.800,0:27:24.300
here have 1/2 * 41

0:27:24.300,0:27:27.485
* 1. Plus

0:27:27.485,0:27:34.790
101. Let me just turn[br]the page over and write this

0:27:34.790,0:27:36.458
some down again.

0:27:37.120,0:27:43.564
SN is equal[br]to 1/2 *

0:27:43.564,0:27:48.934
41 * 1[br]+ 101.

0:27:50.040,0:27:57.432
So we have 1/2 times by 41[br]times by 102 and we can cancel

0:27:57.432,0:28:04.824
it to there to give US 41[br]times by 51. And to do that

0:28:04.824,0:28:10.632
I'd want to get out my[br]Calculator, but we'll leave it

0:28:10.632,0:28:12.744
there to be finished.

0:28:13.270,0:28:16.858
So that's one kind of problem.

0:28:17.800,0:28:21.877
Let's have a look at another[br]kind of problem.

0:28:22.430,0:28:28.214
Let's say we've got an[br]arithmetic progression whose

0:28:28.214,0:28:31.106
first term is 3.

0:28:32.170,0:28:35.530
And the sum.

0:28:36.200,0:28:37.570
Of.

0:28:39.210,0:28:42.970
The first 8.

0:28:44.740,0:28:45.630
Terms.

0:28:47.020,0:28:53.972
Is twice.[br]The sum

0:28:53.972,0:28:59.736
of the[br]first 5

0:28:59.736,0:29:01.177
terms.

0:29:02.570,0:29:04.820
And that seems really quite

0:29:04.820,0:29:09.228
complicated. But it needn't[br]be, but remember this is the

0:29:09.228,0:29:10.374
same arithmetic progression.

0:29:12.250,0:29:18.018
So let's have a think what this[br]is telling us A is equal to

0:29:18.018,0:29:23.374
three and the sum of the first 8[br]terms. Well, to begin with,

0:29:23.374,0:29:28.318
let's write down what the sum of[br]the first 8 terms is.

0:29:28.870,0:29:31.818
Well, it's a half.

0:29:32.440,0:29:39.088
Times N Times[br]2A plus and

0:29:39.088,0:29:41.304
minus 1D.

0:29:42.470,0:29:45.848
And N is equal to 8.

0:29:46.930,0:29:48.750
So we've got a half.

0:29:49.980,0:29:57.630
Times 8. 2A[br]plus N minus one is

0:29:57.630,0:30:05.484
7D. So S 8[br]is equal to half of

0:30:05.484,0:30:11.828
eight is 4 * 2[br]A Plus 7D.

0:30:12.970,0:30:20.434
But we also know that a[br]is equal to three, so we

0:30:20.434,0:30:27.898
can put that in there as[br]well. That's 4 * 6 because

0:30:27.898,0:30:31.630
a is 3 + 7 D.

0:30:32.340,0:30:39.812
Next one, the sum[br]of the first 5

0:30:39.812,0:30:46.834
terms. Let me just write[br]down some of the first

0:30:46.834,0:30:48.862
8 terms were.

0:30:49.150,0:30:55.554
4. Times[br]6 minus

0:30:55.554,0:31:01.146
plus 7D[br]first 5

0:31:01.146,0:31:08.706
terms. Half times the number of[br]terms. That's 5 * 2 A plus

0:31:08.706,0:31:15.888
N minus one times by D will.[br]That must be 4 because any is

0:31:15.888,0:31:17.940
5 times by D.

0:31:18.900,0:31:25.478
So much is 5 over 2 and[br]let's remember that a is equal

0:31:25.478,0:31:32.562
to three, so that 6 + 4[br]D. So I've got S 8 and

0:31:32.562,0:31:39.140
I've got S5 and the question[br]said that S8 was equal to twice

0:31:39.140,0:31:42.970
as five. So I can write this

0:31:42.970,0:31:48.635
for S8. Is[br]equal to

0:31:48.635,0:31:56.030
twice. This which is[br]S five 2 * 5 over two

0:31:56.030,0:32:02.306
6 + 4 D and what seemed[br]a very difficult question as

0:32:02.306,0:32:08.059
reduced itself to an ordinary[br]linear equation in terms of D.

0:32:08.059,0:32:14.335
So we can do some cancelling[br]there and we can multiply out

0:32:14.335,0:32:21.657
the brackets for six is a 24[br]+ 28, D is equal to 56R.

0:32:21.690,0:32:24.974
30 + 5 fours

0:32:24.974,0:32:31.870
are 20D. I can take[br]20D from each side that gives me

0:32:31.870,0:32:33.256
8 D there.

0:32:33.830,0:32:40.998
And I can take 24 from each[br]side, giving me six there. So D

0:32:40.998,0:32:42.534
is equal to.

0:32:43.650,0:32:49.994
Dividing both sides by 8,[br]six over 8 or 3/4 so I know

0:32:49.994,0:32:54.874
everything now that I could[br]possibly want to know about

0:32:54.874,0:32:56.338
this arithmetic progression.

0:32:57.940,0:33:04.142
Now let's go on and have a look[br]at our second type of special

0:33:04.142,0:33:05.471
sequence, a geometric

0:33:05.471,0:33:11.398
progression. So.[br]Take these

0:33:11.398,0:33:14.666
two six

0:33:14.666,0:33:20.680
1854. Let's have a look[br]at how this sequence of numbers

0:33:20.680,0:33:23.207
is growing. We have two. Then we

0:33:23.207,0:33:31.060
have 6. And then we have[br]18. Well 326 and three sixes

0:33:31.060,0:33:38.896
are 18 and three eighteens are[br]54. So this sequence is growing

0:33:38.896,0:33:45.426
by multiplying by three each[br]time. What about this sequence

0:33:45.426,0:33:48.436
one? Minus

0:33:48.436,0:33:52.215
2 four. Minus

0:33:52.215,0:33:57.040
8. What's happening here? We can[br]see the signs are alternating,

0:33:57.040,0:33:58.996
but let's just look at the

0:33:58.996,0:34:05.067
numbers. 1 * 2 would be two 2 *[br]2 would be four. 2 * 4 would be

0:34:05.067,0:34:10.890
8. But if we made that minus[br]two, then one times minus two

0:34:10.890,0:34:17.945
would be minus 2 - 2 times minus[br]two would be plus 4 + 4 times by

0:34:17.945,0:34:23.340
minus two would be minus 8, so[br]this sequence to be generated is

0:34:23.340,0:34:27.905
being multiplied by minus two.[br]Each term is multiplied by minus

0:34:27.905,0:34:30.395
two to give the next term.

0:34:31.210,0:34:36.840
These are examples of geometric[br]progressions, or if you like,

0:34:36.840,0:34:42.830
GPS. Let's try and write one[br]down in general using some

0:34:42.830,0:34:48.810
algebra. So like the AP, we take[br]A to be the first term.

0:34:49.640,0:34:54.249
Now we need something like D.[br]The common difference, but what

0:34:54.249,0:35:00.115
we use is the letter R and we[br]call it the common ratio, and

0:35:00.115,0:35:05.143
that's the number that does the[br]multiplying of each term to give

0:35:05.143,0:35:06.400
the next term.

0:35:07.090,0:35:14.395
So 3 times by two gives us 6,[br]so that's the R. In this case

0:35:14.395,0:35:18.291
the three. So we do a Times by

0:35:18.291,0:35:24.999
R. And then we multiply by, in[br]this case by three again 3 times

0:35:24.999,0:35:29.955
by 6 gives 18, so we multiply by[br]R again, AR squared.

0:35:30.670,0:35:38.122
And then we multiply by three[br]again to give us the 54.

0:35:38.122,0:35:41.848
So by our again AR cubed.

0:35:42.680,0:35:49.638
And what's our end term in this[br]case? While A is the first term

0:35:49.638,0:35:56.596
8 times by R, is the second term[br]8 times by R-squared is the

0:35:56.596,0:36:03.554
third term 8 times by R cubed?[br]Is the fourth term, so it's a

0:36:03.554,0:36:10.015
times by R to the N minus one.[br]Because this power there's a

0:36:10.015,0:36:12.997
one. There is always one less.

0:36:13.000,0:36:17.268
And the number of the term,[br]then its position in the

0:36:17.268,0:36:22.700
sequence. And this is the end[br]term, so it's a Times my R to

0:36:22.700,0:36:24.252
the N minus one.

0:36:25.470,0:36:31.766
What about adding up a[br]geometric progression? Let's

0:36:31.766,0:36:39.636
write that down. SN is[br]equal to a plus R

0:36:39.636,0:36:41.997
Plus R-squared Plus.

0:36:42.580,0:36:50.070
Plus AR to the N minus one,[br]and that's the sum of N terms.

0:36:50.810,0:36:56.342
Going to use another trick[br]similar but not the same to what

0:36:56.342,0:37:00.491
we did with arithmetic[br]progressions. What I'm going to

0:37:00.491,0:37:05.562
do is I'm going to multiply[br]everything by the common ratio.

0:37:06.590,0:37:11.894
So I've multiplied SN by are[br]going to multiply this one by R,

0:37:11.894,0:37:17.198
but I'm not going to write the[br]answer there. I'm going to write

0:37:17.198,0:37:23.726
it here so I've a Times by R and[br]I've written it there plus now I

0:37:23.726,0:37:29.030
multiply this one by R and that[br]would give me a R-squared. I'm

0:37:29.030,0:37:31.070
going to write it there.

0:37:31.630,0:37:36.089
So that term is being multiplied[br]by R and it's gone to their

0:37:36.089,0:37:40.205
that's being multiplied by R and[br]it's gone to their. This one

0:37:40.205,0:37:45.350
will be multiplied by R and it[br]will be a R cubed and it will

0:37:45.350,0:37:46.722
have gone to their.

0:37:47.360,0:37:52.427
Plus etc plus, and we think[br]about what's happening.

0:37:53.350,0:37:58.250
That term will come to here and[br]it will look just like that one.

0:37:59.010,0:38:03.582
Plus and then we need to[br]multiply this by R, and that's

0:38:03.582,0:38:07.011
another. Are that we're[br]multiplying by, so that means

0:38:07.011,0:38:09.297
that becomes AR to the N.

0:38:10.230,0:38:16.548
Now look at why I've lined these[br]up AR, AR, AR squared. Our

0:38:16.548,0:38:19.464
squared, al, cubed, cubed and so

0:38:19.464,0:38:25.423
on. So let's take these two[br]lines of algebra away from each

0:38:25.423,0:38:31.625
other, so I'll have SN minus R[br]times by SN is equal to. Now

0:38:31.625,0:38:38.270
have nothing here to take away[br]from a, so the a stays as it is.

0:38:38.270,0:38:42.700
Then I've AR takeaway are, well,[br]that's nothing. A R-squared

0:38:42.700,0:38:46.244
takeaway R-squared? That's[br]nothing again, same there. And

0:38:46.244,0:38:50.231
so on and so on. AR to the N

0:38:50.231,0:38:55.054
minus one. Take away a art. The[br]end minus one nothing and then

0:38:55.054,0:38:57.532
at the end I have nothing there

0:38:57.532,0:39:02.018
take away.[br]AR to the N.

0:39:03.240,0:39:07.049
Now I need to look closely at[br]both sides of what I've got

0:39:07.049,0:39:10.858
written down, and I'm going to[br]turn this over and write it down

0:39:10.858,0:39:18.516
again. So we've SN minus[br]RSN is equal to A.

0:39:19.020,0:39:22.240
Minus AR to the N.

0:39:22.930,0:39:28.429
Now here I've got a common[br]factor SN the some of the end

0:39:28.429,0:39:34.351
terms when I take that out, I've[br]won their minus R of them there,

0:39:34.351,0:39:41.542
so I get SN times by one minus R[br]is equal 2 and here I've got a

0:39:41.542,0:39:48.310
common Factor A and I can take a[br]out giving me one minus R to the

0:39:48.310,0:39:53.809
N. Remember it was the sum of N[br]terms that I wanted so.

0:39:53.860,0:40:00.466
SN is equal to a Times 1 minus R[br]to the N and to get the SN on

0:40:00.466,0:40:05.604
its own, I've had to divide by[br]one minus R, so I must divide

0:40:05.604,0:40:07.439
this by one minus R.

0:40:09.370,0:40:15.844
And that's my formula for the[br]sum of N terms of a geometric

0:40:15.844,0:40:19.828
progression. And let's just[br]remind ourselves what the

0:40:19.828,0:40:24.808
symbols are N is equal to the[br]number of terms.

0:40:24.820,0:40:30.847
A is the first[br]term of our

0:40:30.847,0:40:34.291
geometric[br]progression and are

0:40:34.291,0:40:40.318
we said was called[br]the common ratio.

0:40:41.420,0:40:48.011
OK, and let's just remember the[br]NTH term in the sequence was AR

0:40:48.011,0:40:55.109
to the N minus one. So those[br]are our fax so far about GPS

0:40:55.109,0:41:00.686
or geometric progressions. Let's[br]see if we can use these facts

0:41:00.686,0:41:07.277
in order to be able to help[br]us solve some problems and do

0:41:07.277,0:41:15.027
some questions. So first of all,[br]let's take this 2 + 6 +

0:41:15.027,0:41:22.268
18 + 54 plus. Let's say there[br]are six terms. What's the answer

0:41:22.268,0:41:25.610
when it comes to adding those

0:41:25.610,0:41:32.550
up? Well, we know that a[br]is equal to two. We know that

0:41:32.550,0:41:39.825
our is equal to three and we[br]know that N is equal to six. So

0:41:39.825,0:41:47.585
to solve that, all we need to do[br]is write down that the sum of N

0:41:47.585,0:41:54.860
terms is a Times 1 minus R to[br]the N all over 1 minus R.

0:41:54.860,0:41:57.285
Substitute our numbers in two

0:41:57.285,0:42:03.904
times. 1 - 3[br]to the power 6.

0:42:04.520,0:42:07.991
Over 1 -

0:42:07.991,0:42:14.612
3. So this is 2 * 1 -[br]3 to the power six over minus

0:42:14.612,0:42:20.044
two, and we can cancel a minus[br]two with the two that we leave

0:42:20.044,0:42:24.700
as with a minus one there and[br]one there if I multiply

0:42:24.700,0:42:29.744
throughout by the minus one,[br]I'll have minus 1 * 1 is minus

0:42:29.744,0:42:35.952
one and minus one times minus 3[br]to the six is 3 to the 6th, so

0:42:35.952,0:42:39.444
the sum of N terms is 3 to the

0:42:39.444,0:42:45.430
power 6. Minus one and with a[br]Calculator we could workout what

0:42:45.430,0:42:49.056
3 to the power 6 - 1

0:42:49.056,0:42:52.256
was. Let's take

0:42:52.256,0:42:59.440
another. Question to do with[br]summing the terms of a geometric

0:42:59.440,0:43:05.892
progression. What's the sum[br]of that? Let's say for five

0:43:05.892,0:43:11.772
terms. While we can begin by[br]identifying the first term,

0:43:11.772,0:43:15.888
that's eight, and what's the[br]common ratio?

0:43:17.100,0:43:23.100
Well, to go from 8 to 4 as a[br]number we would have it, but

0:43:23.100,0:43:27.900
there's a minus sign in there.[br]So that suggests that the common

0:43:27.900,0:43:32.700
ratio is minus 1/2. Let's just[br]check it minus four times. By

0:43:32.700,0:43:39.100
minus 1/2 is plus 2 + 2 times Y[br]minus 1/2 is minus one, and we

0:43:39.100,0:43:41.900
said five terms, so Ann is equal

0:43:41.900,0:43:49.494
to 5. So we can write[br]down our formula. SN is equal to

0:43:49.494,0:43:57.390
a Times 1 minus R to the[br]power N all over 1 minus R.

0:43:57.980,0:44:01.120
And so A is 8.

0:44:02.560,0:44:10.220
1 minus and this is[br]minus 1/2 to the power

0:44:10.220,0:44:16.460
5. All over 1 minus minus[br]1/2. You can see these

0:44:16.460,0:44:19.328
questions get quite[br]complicated with the

0:44:19.328,0:44:24.108
arithmetic, so you have to[br]be very careful and you

0:44:24.108,0:44:27.932
have to have a good[br]knowledge of fractions.

0:44:29.320,0:44:37.112
This is 8 * 1. Now let's have[br]a look at minus 1/2 to the power

0:44:37.112,0:44:42.360
5. Well, I'm multiplying the[br]minus sign by itself five times,

0:44:42.360,0:44:47.136
which would give me a negative[br]number, and I've got a minus

0:44:47.136,0:44:51.912
sign there outside the bracket.[br]That's going to mean I've got 6

0:44:51.912,0:44:57.086
minus signs together. Makes it[br]plus. So now I can look at the

0:44:57.086,0:44:59.076
half to the power 5.

0:44:59.670,0:45:02.574
Well, that's going to be one

0:45:02.574,0:45:09.270
over. 248-1632 to[br]to the power

0:45:09.270,0:45:12.696
five is 32.

0:45:13.400,0:45:20.504
All over 1 minus minus 1/2.[br]That's 1 + 1/2. Let's write

0:45:20.504,0:45:23.464
that as three over 2.

0:45:24.540,0:45:27.170
So this is equal to.

0:45:28.160,0:45:30.700
Now I've got 8.

0:45:31.450,0:45:37.764
Times by one plus, one over 32,[br]and I'm dividing by three over 2

0:45:37.764,0:45:42.725
to divide by a fraction. We[br]invert the fraction that's two

0:45:42.725,0:45:49.039
over 3 and we multiply by and we[br]just turn the page to finish

0:45:49.039,0:45:50.392
this one off.

0:45:51.110,0:45:58.840
So we have SN is[br]equal to 8 * 1

0:45:58.840,0:46:06.570
+ 1 over 32 times[br]by 2/3 is equal to

0:46:06.570,0:46:14.300
8 times by now one[br]and 132nd. Well, there are

0:46:14.300,0:46:22.030
3230 seconds in one, so[br]altogether there I've got 33.

0:46:22.030,0:46:25.264
30 seconds times

0:46:25.264,0:46:30.926
by 2/3. And we[br]can do some canceling threes

0:46:30.926,0:46:35.582
into 30. Three will go 11 and[br]threes into three. There goes

0:46:35.582,0:46:42.536
one. Twos into two goes one and[br]tools into 32, goes 16 and 18

0:46:42.536,0:46:48.864
two eight goes one and eight[br]into 16 goes 2. So we 1 *

0:46:48.864,0:46:55.192
11 * 1 that's just 11 over 2[br]because we've 2 times by one

0:46:55.192,0:47:01.068
there. So we love Nova two or we[br]prefer five and a half.

0:47:01.840,0:47:07.216
So that we've got the some of[br]those five terms of that

0:47:07.216,0:47:12.144
particular GP. Five and a half,[br]11 over 2 or 5.5.

0:47:12.680,0:47:19.775
But here's a different question.[br]What if we've got the sequence

0:47:19.775,0:47:27.512
248? 128 how many terms are[br]we got? How many bits do we

0:47:27.512,0:47:33.944
need to get from 2 up to[br]128? Well, let's begin by

0:47:33.944,0:47:36.088
identifying the first term

0:47:36.088,0:47:39.610
that's two. This is.

0:47:40.820,0:47:45.792
A geometric progression because[br]we multiply by two to get each

0:47:45.792,0:47:52.120
term. So the common ratio are is[br]2 and what we don't know is

0:47:52.120,0:47:59.168
what's N. So let's have a look.[br]This is the last term and we

0:47:59.168,0:48:05.524
know our expression for the last[br]term. 128 is equal to AR to the

0:48:05.524,0:48:06.886
N minus one.

0:48:07.700,0:48:14.427
So let's substituting some[br]of our information.

0:48:14.450,0:48:22.310
A is 2 times by two[br]4R to the N minus one.

0:48:23.010,0:48:29.020
Well, we can divide both[br]sides by this two here,

0:48:29.020,0:48:31.424
which will give us.

0:48:32.570,0:48:38.240
64 is equal to two to the[br]N minus one.

0:48:38.870,0:48:45.170
I think about that it's 248[br]sixteen 3264 so I had to

0:48:45.170,0:48:51.995
multiply 2 by itself six times[br]in order to get 64, so 2

0:48:51.995,0:48:59.345
to the power 6, which is 64[br]is equal to 2 to the power

0:48:59.345,0:49:06.695
N minus one, so six is equal[br]to N minus one, and so N

0:49:06.695,0:49:09.845
is equal to 7, adding one.

0:49:09.860,0:49:14.960
To each side. In other words,[br]there were Seven terms in our.

0:49:15.520,0:49:22.122
Geometric progression. Type[br]of question that's often given

0:49:22.122,0:49:26.698
for geometric progressions is[br]given a geometric progression.

0:49:26.698,0:49:32.990
How many terms do you need[br]to add together before you

0:49:32.990,0:49:38.710
exceed a certain limit? So, for[br]instance, here's a geometric.

0:49:38.820,0:49:44.832
Progression. How many times of[br]this geometric progression do we

0:49:44.832,0:49:50.684
need to act together in order to[br]be sure that the some of them

0:49:50.684,0:49:52.356
will get over 20?

0:49:53.040,0:49:58.932
Well, first of all, let's try[br]and identify this as a geometric

0:49:58.932,0:50:04.333
progression. The first term is[br]on and it looks like what's

0:50:04.333,0:50:09.243
doing the multiplying. The[br]common ratio is 1.1. Let's just

0:50:09.243,0:50:10.716
check that here.

0:50:11.290,0:50:16.919
1.1 times by one point, one[br]well. That's kind of like 11 *

0:50:16.919,0:50:18.218
11 is 121.

0:50:18.730,0:50:23.086
With two numbers after the[br]decimal point in one point 1 *

0:50:23.086,0:50:27.442
1.1 and with two numbers after[br]the decimal point there. So yes,

0:50:27.442,0:50:29.257
this is a geometric progression.

0:50:30.050,0:50:36.719
So let's write down our formula[br]for N terms sum of N terms

0:50:36.719,0:50:42.362
is equal to a Times 1 minus[br]R to the N.

0:50:42.970,0:50:50.320
All over 1 minus R. We want to[br]know what value of N is just

0:50:50.320,0:50:53.260
going to take us over 20.

0:50:53.880,0:51:00.090
So let's substituting some[br]numbers. This is one for

0:51:00.090,0:51:03.540
a 1 - 1.1 to

0:51:03.540,0:51:10.816
the N. All over[br]1 - 1.1 that

0:51:10.816,0:51:15.952
has to be greater[br]than 20.

0:51:17.210,0:51:22.886
So one times by that isn't going[br]to affect what's in the

0:51:22.886,0:51:29.981
brackets. That would be 1 - 1.1[br]to the N all over 1 - 1.1

0:51:29.981,0:51:35.184
is minus nought. .1 that has to[br]be greater than 20.

0:51:36.510,0:51:42.256
Now if I use the minus sign[br]wisely. In other words, If I

0:51:42.256,0:51:44.024
divide if you like.

0:51:45.280,0:51:48.724
Minus note .1 into there as

0:51:48.724,0:51:52.110
a. Division, then I'll have.

0:51:52.690,0:51:58.878
The minus sign will make that a[br]minus and make that a plus, so

0:51:58.878,0:52:05.066
I'll have one point 1 to the N[br]minus one and divided by North

0:52:05.066,0:52:10.370
Point one is exactly the same as[br]multiplying by 10. That means

0:52:10.370,0:52:12.580
I've got a 10 here.

0:52:13.310,0:52:16.467
That I can divide both sides by.

0:52:17.620,0:52:23.613
So let's just write this down[br]again 1.1 to the N minus one

0:52:23.613,0:52:29.606
times by 10 has to be greater[br]than 20. So let's divide both

0:52:29.606,0:52:36.521
sides by 10, one point 1 to the[br]N minus one has to be greater

0:52:36.521,0:52:43.897
than two and will add the one to[br]both sides 1.1 to the end has to

0:52:43.897,0:52:45.741
be greater than three.

0:52:46.550,0:52:52.010
Problem how do we find N? One of[br]the ways of solving equations

0:52:52.010,0:52:57.470
like this is to take logarithms[br]of both sides, so I'm going to

0:52:57.470,0:53:02.510
take natural logarithms of both[br]sides. I'm going to do it to

0:53:02.510,0:53:07.130
this site first. That's the[br]natural logarithm of 3 N about

0:53:07.130,0:53:12.590
this side. When you're taking a[br]log of a number that's raised to

0:53:12.590,0:53:17.210
the power, that's the equivalent[br]of multiplying the log of that

0:53:17.210,0:53:23.191
number. By the power that's N[br]times the log of 1.1. Well

0:53:23.191,0:53:29.704
now this is just an equation[br]for N because N has got to

0:53:29.704,0:53:36.217
be greater than the log of 3[br]divided by the log of 1.1

0:53:36.217,0:53:37.720
because after all.

0:53:39.200,0:53:44.645
Log of three is just a number[br]and log of 1.1 is just a number

0:53:44.645,0:53:48.275
and this is the sort of[br]calculation that really does

0:53:48.275,0:53:50.816
have to be done on a Calculator.

0:53:51.800,0:53:57.146
So if we take our Calculator and[br]we turn it on.

0:53:58.360,0:54:02.370
And we do the calculation. The[br]natural log of three.

0:54:03.930,0:54:09.678
Divided by the natural log of[br]1.1, we ask our Calculator to

0:54:09.678,0:54:16.384
calculate that for us. It tells[br]us that it's 11.5 to 6 and some

0:54:16.384,0:54:20.695
more decimal places. We're not[br]really worried about these

0:54:20.695,0:54:26.922
decimal places. An is a whole[br]number and it has to be greater

0:54:26.922,0:54:33.628
than 11 and some bits, so N has[br]got to be 12 or more.

0:54:35.970,0:54:43.458
That's one last twist[br]to our geometric progression.

0:54:43.460,0:54:46.900
Let's have a look at

0:54:46.900,0:54:53.825
this one. What have[br]we got got

0:54:53.825,0:55:00.479
a geometric progression.[br]First term a

0:55:00.479,0:55:02.697
is one.

0:55:04.360,0:55:10.910
Common ratio is 1/2 because[br]we're multiplying by 1/2 each

0:55:10.910,0:55:14.800
time. That write down

0:55:14.800,0:55:21.456
some sums. S1, the sum[br]of the first term is just.

0:55:22.060,0:55:25.345
1. What's

0:55:25.345,0:55:32.665
S2? That's the[br]sum of the first 2 terms, so

0:55:32.665,0:55:37.070
that's. Three over[br]2.

0:55:38.460,0:55:43.420
What's the sum of the first[br]three terms? That's one.

0:55:44.560,0:55:48.106
Plus 1/2 +

0:55:48.106,0:55:55.781
1/4. Add those up[br]in terms of how many quarters

0:55:55.781,0:55:59.555
have we got then that is

0:55:59.555,0:56:06.695
7. Quarters As[br]for the sum of

0:56:06.695,0:56:08.505
the first.

0:56:08.760,0:56:15.339
4. Terms.[br]Add those up in terms of how

0:56:15.339,0:56:21.037
many eighths if we got so we've[br]got eight of them there. Four of

0:56:21.037,0:56:26.328
them there. That's 12. Two of[br]them there. That's 14 and one of

0:56:26.328,0:56:28.363
them there. That's 15 eighths.

0:56:29.090,0:56:31.978
Seems to be some sort of[br]pattern here.

0:56:32.990,0:56:36.374
Here we seem to be 1/2 short of

0:56:36.374,0:56:42.690
two. Here we[br]seem to

0:56:42.690,0:56:49.410
be 1/4. Short of two[br]here, we seem to be an eighth

0:56:49.410,0:56:55.052
short of two and we look at the[br]first one. Then we're clearly 1

0:56:55.052,0:56:56.261
short of two.

0:56:56.930,0:57:04.014
He's a powers of two. Let's have[br]a look 2 - 2 to the

0:57:04.014,0:57:09.580
power zero, 'cause 2 to the[br]power zero is 1 two.

0:57:10.310,0:57:11.510
Minus.

0:57:12.750,0:57:20.660
2 to the power minus one 2[br]- 2 to the power minus two

0:57:20.660,0:57:24.615
2 - 2 to the power minus

0:57:24.615,0:57:29.414
three. But each of these is[br]getting smaller. We're getting

0:57:29.414,0:57:34.538
nearer and nearer to two. The[br]next one we take away will be a

0:57:34.538,0:57:40.028
16th, the one after that will be[br]a 32nd and the next bit we take

0:57:40.028,0:57:45.884
off 2 is going to be a 64th and[br]then a 128 and then at one

0:57:45.884,0:57:50.276
256th. So we're getting the bits[br]were taking away from two are

0:57:50.276,0:57:53.570
getting smaller and smaller and[br]smaller until eventually we

0:57:53.570,0:57:56.498
wouldn't be able to distinguish[br]them from zero.

0:57:56.510,0:58:01.592
And so if we could Add all of[br]these up forever, a sum to

0:58:01.592,0:58:06.674
Infinity, if you like the[br]answer, or to be 2 or as near as

0:58:06.674,0:58:12.482
we want to be to two. So let's[br]see if we can have a look at

0:58:12.482,0:58:13.934
that with some algebra.

0:58:14.760,0:58:22.208
We know that the sum to end[br]terms is equal to a Times 1

0:58:22.208,0:58:27.528
minus R to the N all over[br]1 minus R.

0:58:28.050,0:58:33.118
What we want to have a look at[br]is this thing are because what

0:58:33.118,0:58:34.566
was crucial about this?

0:58:35.960,0:58:42.912
Geometric progression was at the[br]common ratio was a half a

0:58:42.912,0:58:45.440
number less than one.

0:58:45.450,0:58:48.192
So let's have a look what

0:58:48.192,0:58:55.928
happens. When all is bigger than[br]one to R to the power N.

0:58:56.630,0:59:01.778
We are is bigger than one and we[br]keep multiplying it by itself.

0:59:02.280,0:59:08.737
Grows, it grows very rapidly and[br]really gets very big very

0:59:08.737,0:59:15.781
quickly. Check it with two, 2,[br]four, 816. It goes off til

0:59:15.781,0:59:20.880
Infinity. And because it goes[br]off to Infinity, it takes the

0:59:20.880,0:59:22.875
sum with it as well.

0:59:24.350,0:59:28.232
What about if our is equal

0:59:28.232,0:59:34.380
to 1? Well, we can't really[br]use this formula then because we

0:59:34.380,0:59:39.024
would be dividing by zero. But[br]if you think about it, are

0:59:39.024,0:59:44.442
equals 1 means every term is the[br]same. So if we start off with

0:59:44.442,0:59:49.860
one every term is the same 1111[br]and you just add them all up.

0:59:49.860,0:59:55.278
But again that means the sum is[br]going to go off to Infinity if

0:59:55.278,0:59:59.535
you take the number any number[br]and add it to itself.

1:00:00.340,1:00:04.504
An infinite number of times[br]you're going to get a very, very

1:00:04.504,1:00:12.083
big number. What happens if[br]our is less than minus one?

1:00:12.083,1:00:14.711
Something like minus 2?

1:00:15.450,1:00:19.751
Well, what's going to happen[br]then to R to the N?

1:00:20.920,1:00:26.058
Well, it's going to be plus an.[br]It's going to be minus as we

1:00:26.058,1:00:31.196
multiply by this number such as[br]minus two. So we have minus 2 +

1:00:31.196,1:00:35.600
4 minus A. The thing to notice[br]is it's getting bigger, it's

1:00:35.600,1:00:40.371
getting bigger each time. So[br]again are to the end is going to

1:00:40.371,1:00:44.041
go off to Infinity. It's going[br]to oscillate between plus

1:00:44.041,1:00:48.445
Infinity and minus Infinity, but[br]it's going to get very big and

1:00:48.445,1:00:51.748
that means this sum is also[br]going to get.

1:00:51.750,1:00:52.490
Very big.

1:00:53.570,1:01:00.086
What about our equals minus one?[br]Well, if R equals minus one,

1:01:00.086,1:01:05.516
let's think about a sequence[br]like that. Well, a typical

1:01:05.516,1:01:08.774
sequence might be 1 - 1.

1:01:09.490,1:01:14.092
1 - 1 and we can see the[br]problem. It depends where we

1:01:14.092,1:01:19.402
stop. If I stop here the sum is[br]0 but if I put another one

1:01:19.402,1:01:24.004
there, the sum is one. So we've[br]got an infinite number of terms

1:01:24.004,1:01:28.606
then. Well, it depends on money[br]I've got us to what the answer

1:01:28.606,1:01:33.562
is so there isn't a limit for[br]SN. There isn't a thing that it

1:01:33.562,1:01:35.686
can come to a definite number.

1:01:36.670,1:01:40.963
Let's have a look. We've[br]considered all possible values

1:01:40.963,1:01:46.687
of our except those where are is[br]between plus and minus one.

1:01:47.230,1:01:50.270
Let's take our equals 1/2[br]as an example.

1:01:51.590,1:01:54.446
Or half trans by half is 1/4.

1:01:55.470,1:01:59.208
Reply by 1/2. Again[br]that's an eighth.

1:02:00.410,1:02:03.170
Multiply by 1/2 again, that's a

1:02:03.170,1:02:08.846
16. Multiplied by 1/2 again,[br]that's a 32nd.

1:02:09.640,1:02:14.810
By half again that's a 64th by[br]1/2 again, that's 128.

1:02:16.080,1:02:21.816
It's getting smaller, and if we[br]do it enough times then it's

1:02:21.816,1:02:24.684
going to head off till 0.

1:02:25.690,1:02:30.530
What about a negative one? You[br]might say, let's think about

1:02:30.530,1:02:36.176
minus 1/2. Now multiplied by[br]minus 1/2, it's a quarter.

1:02:36.176,1:02:41.496
Multiply the quarter by minus[br]1/2. It's minus an eighth.

1:02:41.496,1:02:46.816
Multiply again by minus 1/2.[br]Well, that's plus a 16th.

1:02:46.816,1:02:52.668
Multiply again by minus 1/2.[br]That's minus a 32nd, so we're

1:02:52.668,1:02:59.052
approaching 0, but where dotting[br]about either side of 0 plus them

1:02:59.052,1:03:03.840
were minus, then were plus then[br]where mine is.

1:03:03.870,1:03:08.562
We're getting nearer to zero[br]each time, so again are to the

1:03:08.562,1:03:13.645
power. N is going off to zero.[br]What does that mean? It means

1:03:13.645,1:03:19.119
that this some. Here we can have[br]what we call a sum to Infinity.

1:03:19.119,1:03:23.811
Sometimes it's just written with[br]an S and sometimes it's got a

1:03:23.811,1:03:25.766
little Infinity sign on it.

1:03:26.480,1:03:32.822
What that tells us? Because this[br]art of the end is going off to

1:03:32.822,1:03:40.070
0 then it's a times by one over[br]1 minus R and that's our sum to

1:03:40.070,1:03:42.788
Infinity. In other words, we can

1:03:42.788,1:03:48.934
add up. An infinite number of[br]terms for a geometric

1:03:48.934,1:03:55.772
progression provided. The common[br]ratio is between one and minus

1:03:55.772,1:04:02.801
one, so let's have a look[br]at an example. Supposing we've

1:04:02.801,1:04:04.718
got this row.

1:04:04.750,1:04:12.120
Metric progression.[br]Well, first term is one

1:04:12.120,1:04:16.524
now a common ratio is[br]1/3.

1:04:17.710,1:04:19.796
And what does this come to when

1:04:19.796,1:04:26.460
we add up? As many terms as[br]we can, what's the sum to

1:04:26.460,1:04:32.652
Infinity? We know the formula[br]that's a over 1 minus R, so

1:04:32.652,1:04:39.360
let's put the numbers in this[br]one for a over 1 - 1/3.

1:04:40.460,1:04:46.323
So the one on tops OK and the[br]one minus third. Well that's

1:04:46.323,1:04:52.186
2/3, and if we're dividing by a[br]fraction then we invert it and

1:04:52.186,1:04:57.147
multiply. So altogether that[br]would come to three over 2, so

1:04:57.147,1:04:59.853
it's very easy formula to use.

1:05:01.510,1:05:04.246
Finally, just let's recap for a

1:05:04.246,1:05:06.710
geometric progression. A.

1:05:07.360,1:05:10.760
Is the first term.

1:05:10.760,1:05:13.480
Aw.

1:05:14.740,1:05:16.309
Is the common.

1:05:17.140,1:05:24.026
Ratio. So a[br]geometric progression looks like

1:05:24.026,1:05:31.436
AARA, R-squared, AR, cubed and[br]the N Terminus series AR

1:05:31.436,1:05:35.141
to the N minus one.

1:05:35.780,1:05:41.060
And if we want to add up this[br]sequence of numbers SN.

1:05:41.820,1:05:49.628
Then that's a Times 1 minus R to[br]the power N or over 1 minus R.

1:05:50.490,1:05:55.352
And if we're lucky enough to[br]have our between plus and

1:05:55.352,1:05:59.772
minus one, sometimes that's[br]written as the modulus of art

1:05:59.772,1:06:04.634
is less than one. If we're[br]lucky to have this condition,

1:06:04.634,1:06:10.380
then we can get a sum to[br]Infinity, which is a over 1

1:06:10.380,1:06:11.264
minus R.