WEBVTT 00:00:00.500 --> 00:00:06.560 In this video, we're going to be looking at sequences and series, 00:00:06.560 --> 00:00:11.105 so let's begin by looking at what a sequences. 00:00:12.030 --> 00:00:15.885 This, for instance is a 00:00:15.885 --> 00:00:23.280 sequence. It's a set of numbers. 00:00:23.830 --> 00:00:30.745 And here we seem to have a rule. All of these are odd numbers, or 00:00:30.745 --> 00:00:36.277 we can look at it. We increase by two each time 13579. 00:00:37.280 --> 00:00:42.460 So there's our sequence of odd numbers. 00:00:43.900 --> 00:00:45.788 Here is another sequence. 00:00:46.650 --> 00:00:53.742 These numbers are the 00:00:53.742 --> 00:01:01.102 square numbers. 1 squared, 2 squared, is 4 three 00:01:01.102 --> 00:01:07.066 squared is 9. Four squared is 16 and 5 squared is 25. So again 00:01:07.066 --> 00:01:12.604 we've got a sequence of numbers. We've got a rule that seems to 00:01:12.604 --> 00:01:15.269 produce them. Those are the. 00:01:15.890 --> 00:01:22.140 Square numbers. Is a slightly different sequence. 00:01:22.510 --> 00:01:27.450 Here we've got alternation between one and minus one back 00:01:27.450 --> 00:01:34.366 to one on again to minus one back to one on again to minus 00:01:34.366 --> 00:01:38.812 one. But this is still a sequence of numbers. 00:01:39.510 --> 00:01:42.966 Now, because I've written some dots after it here. 00:01:43.660 --> 00:01:48.769 This means that this is meant to be an infinite sequence. It goes 00:01:48.769 --> 00:01:53.485 on forever and this is meant to be an infinite sequence. It 00:01:53.485 --> 00:01:58.594 carries on forever, and this one does too. If I want a finite 00:01:58.594 --> 00:02:05.570 sequence. What might a finite sequence 00:02:05.570 --> 00:02:09.188 look like, for 00:02:09.188 --> 00:02:15.904 instance 1359? That would be a finite sequence. We've got 00:02:15.904 --> 00:02:18.652 4 numbers and then it stops 00:02:18.652 --> 00:02:25.160 dead. Perhaps if we look at the sequence of square 00:02:25.160 --> 00:02:30.960 numbers 1, four, 916, that again is a finite sequence. 00:02:32.070 --> 00:02:39.950 Sequence that will be very interested in is the sequence 00:02:39.950 --> 00:02:47.042 of whole numbers, the counting numbers, the integers. So 00:02:47.042 --> 00:02:54.922 there's a sequence of integers, and it's finite because it 00:02:54.922 --> 00:03:02.802 stops at N, so we're counting 123456789 up to N. 00:03:02.820 --> 00:03:08.150 And the length of this sequence is an the integer, the number N. 00:03:10.310 --> 00:03:16.277 Very popular sequence of numbers. Quite well known is 00:03:16.277 --> 00:03:18.266 this particular sequence. 00:03:19.000 --> 00:03:23.610 This is a slightly different sequence. It's infinite keeps on 00:03:23.610 --> 00:03:25.915 going and it's called the 00:03:25.915 --> 00:03:30.714 Fibonacci sequence. And we can see how it's generated. This 00:03:30.714 --> 00:03:35.823 number 2 is formed by adding the one and the one together, and 00:03:35.823 --> 00:03:40.539 then the three is formed by adding the one and the two 00:03:40.539 --> 00:03:45.255 together. The Five is formed by adding the two and the three 00:03:45.255 --> 00:03:46.827 together, and so on. 00:03:47.840 --> 00:03:52.585 So there's a question here. How could we write this rule down in 00:03:52.585 --> 00:03:56.600 general, where we can say it that any particular term is 00:03:56.600 --> 00:04:00.615 generated by adding the two numbers that come before it in 00:04:00.615 --> 00:04:04.630 the sequence together? But how might we set that down? How 00:04:04.630 --> 00:04:09.740 might we label it? One way might be to use algebra and say will 00:04:09.740 --> 00:04:14.120 call the first term you, and because it's the first time we 00:04:14.120 --> 00:04:17.040 want to label it, so we call it 00:04:17.040 --> 00:04:20.690 you one. And then the next terminal sequence, the second 00:04:20.690 --> 00:04:25.305 term. It would make sense to call it you too, and the third 00:04:25.305 --> 00:04:29.565 term in our sequence. It would make sense therefore, to call it 00:04:29.565 --> 00:04:36.916 you 3. You four and so on up to UN. So this represents a 00:04:36.916 --> 00:04:43.012 finite sequence that's got N terms in it. If we look at 00:04:43.012 --> 00:04:48.600 the Fibonacci sequence as an example of making use of this 00:04:48.600 --> 00:04:54.696 kind of notation, we could say that the end term UN was 00:04:54.696 --> 00:04:59.776 generated by adding together the two terms that come immediately 00:04:59.776 --> 00:05:01.300 before it will. 00:05:01.320 --> 00:05:05.110 Term that comes immediately before this must have a number 00:05:05.110 --> 00:05:10.416 attached to it. That's one less than N and that would be N minus 00:05:10.416 --> 00:05:15.910 one. Plus on the term that's down, the term that comes 00:05:15.910 --> 00:05:20.926 immediately before this one must have a number attached to it. 00:05:20.926 --> 00:05:26.854 That's one less than that. Well, that's UN minus 1 - 1 taking 00:05:26.854 --> 00:05:30.046 away 2 ones were taking away two 00:05:30.046 --> 00:05:36.642 altogether. So that we can see how we might use the algebra 00:05:36.642 --> 00:05:42.546 this algebraic notation help us write down a rule for the Fibo 00:05:42.546 --> 00:05:46.406 Nachi sequence. OK, how can 00:05:46.406 --> 00:05:52.185 we? Use this in a slightly different way. 00:05:52.710 --> 00:06:00.270 What we need to look at now is to move on and have a 00:06:00.270 --> 00:06:04.050 look what we mean by a series. 00:06:04.630 --> 00:06:11.716 This is a sequence, label it. 00:06:12.490 --> 00:06:17.330 A sequence it's a list of numbers generated by some 00:06:17.330 --> 00:06:22.170 particular rule. It's finite because there are any of them. 00:06:22.900 --> 00:06:28.510 What then, is a series series is what we get. 00:06:29.280 --> 00:06:32.970 When we add. 00:06:32.970 --> 00:06:36.478 Terms of the sequence. 00:06:37.070 --> 00:06:44.078 Together And because we're adding together and terms 00:06:44.078 --> 00:06:46.934 will call this SN. 00:06:47.860 --> 00:06:55.610 The sum of N terms and it's that which is 00:06:55.610 --> 00:06:57.160 the series. 00:06:59.040 --> 00:07:06.097 So. Let's have a look at the sequence 00:07:06.097 --> 00:07:12.825 of numbers 123456, and so on up to 00:07:12.825 --> 00:07:13.666 N. 00:07:15.080 --> 00:07:17.190 Then S1. 00:07:18.850 --> 00:07:22.210 Is just one. 00:07:22.840 --> 00:07:24.700 S2. 00:07:25.750 --> 00:07:32.660 Is the sum of the first 2 terms 1 + 00:07:32.660 --> 00:07:34.980 2? And that gives us 3. 00:07:35.740 --> 00:07:42.893 S3. Is the sum of the first three 00:07:42.893 --> 00:07:46.607 terms 1 + 2 + 3? 00:07:47.220 --> 00:07:50.805 And that gives us 6 00:07:50.805 --> 00:07:58.250 and S4. Is the sum of the first four terms 1 + 00:07:58.250 --> 00:08:02.100 2 + 3 + 4 and that 00:08:02.100 --> 00:08:03.580 gives us. Hey. 00:08:04.480 --> 00:08:10.148 So this gives us the basic vocabulary to be able to move on 00:08:10.148 --> 00:08:14.944 to the next section of the video, but just let's remind 00:08:14.944 --> 00:08:16.688 ourselves first of all. 00:08:17.220 --> 00:08:18.640 A sequence. 00:08:20.180 --> 00:08:24.203 Is a set of numbers generated by some rule. 00:08:25.830 --> 00:08:31.402 A series is what we get when we add the terms of the sequence 00:08:31.402 --> 00:08:37.516 together. This particular sequence has N terms in it 00:08:37.516 --> 00:08:43.486 because we've labeled each term in the sequence with accounting 00:08:43.486 --> 00:08:49.456 number. If you like U1U2U free, you fall you N. 00:08:51.380 --> 00:08:55.921 Now. With this vocabulary of sequences and series in mind, 00:08:55.921 --> 00:09:01.756 we're going to go on and have a look at a 2 special kinds of 00:09:01.756 --> 00:09:06.630 sequences. The first one is called an arithmetic progression 00:09:06.630 --> 00:09:09.990 and the second one is called a 00:09:09.990 --> 00:09:15.125 geometric progression. Will begin with an arithmetic 00:09:15.125 --> 00:09:22.062 progression. Let's start by having a look at this 00:09:22.062 --> 00:09:24.950 sequence of. Odd 00:09:25.470 --> 00:09:32.748 Numbers that we had before 1357. 00:09:33.300 --> 00:09:40.026 Is another sequence not 1020 thirty, 00:09:40.026 --> 00:09:43.389 and so on. 00:09:44.640 --> 00:09:51.634 What we can see in this first sequence is that each term after 00:09:51.634 --> 00:09:56.476 the first one is formed by adding on to. 00:09:57.210 --> 00:10:00.258 1 + 2 gives us 3. 00:10:00.840 --> 00:10:03.858 3 + 2 gives us 5. 00:10:04.710 --> 00:10:11.145 5 + 2 gives us 7 and it's because we're adding on the 00:10:11.145 --> 00:10:17.085 same amount every time. This is an example of what we call 00:10:17.085 --> 00:10:18.075 an arithmetic. 00:10:19.870 --> 00:10:28.082 Progression. If we look at this sequence of 00:10:28.082 --> 00:10:33.538 numbers, we can see exactly the same property we've started with 00:10:33.538 --> 00:10:39.986 zero. We've added on 10, and we've added on 10 again to get 00:10:39.986 --> 00:10:46.930 20. We've had it on 10 again to get 30, so again, this is 00:10:46.930 --> 00:10:49.410 exactly the same. It's an 00:10:49.410 --> 00:10:56.166 arithmetic progression. We don't have to add on things, so 00:10:56.166 --> 00:11:03.283 for instance a sequence of numbers that went like this 8 00:11:03.283 --> 00:11:05.871 five, 2 - 1. 00:11:06.560 --> 00:11:08.665 Minus 00:11:08.665 --> 00:11:14.253 4. If we look what's happening where going from 8:00 00:11:14.253 --> 00:11:18.510 to 5:00, so that's takeaway three were going from five to 00:11:18.510 --> 00:11:23.154 two, so that's takeaway. Three were going from 2 to minus. One 00:11:23.154 --> 00:11:27.411 takeaway. Three were going from minus one to minus four takeaway 00:11:27.411 --> 00:11:33.662 3. Another way of thinking about takeaway three is to say where 00:11:33.662 --> 00:11:35.510 adding on minus three. 00:11:36.330 --> 00:11:43.050 8 at minus three is 5 five at minus three is 2, two AD minus 00:11:43.050 --> 00:11:48.426 three is minus one, so again, this is an example of an 00:11:48.426 --> 00:11:54.084 arithmetic progression. And what we want to be able to do is to 00:11:54.084 --> 00:11:56.996 try and encapsulate this arithmetic progression in some 00:11:56.996 --> 00:11:59.180 algebra, so we'll use the letter 00:11:59.180 --> 00:12:04.686 A. To stand for the first term. 00:12:05.310 --> 00:12:12.360 And will use the letter D to stand for the 00:12:12.360 --> 00:12:17.768 common difference. Now the common difference is the 00:12:17.768 --> 00:12:23.686 difference between each term and it's called common because it is 00:12:23.686 --> 00:12:30.142 common to each between each term. So let's have a look at 00:12:30.142 --> 00:12:36.598 one 357 and let's have a think about how it's structured 13. 00:12:37.340 --> 00:12:44.820 5. 7 and so on. So we begin with one and 00:12:44.820 --> 00:12:48.817 then the three is 1 + 2. 00:12:50.270 --> 00:12:57.755 The Five is 1 + 2 tools because by the time we got to five, 00:12:57.755 --> 00:13:00.749 we've added four onto the one. 00:13:01.420 --> 00:13:07.894 The Seven is one plus. Now the time we've got to Seven, we've 00:13:07.894 --> 00:13:14.368 added three tools on. Let's just do one more. Let's put nine in 00:13:14.368 --> 00:13:19.846 there and that would be 1 + 4 times by two. 00:13:20.980 --> 00:13:25.320 So let's see if we can begin to write this down. This is one. 00:13:26.280 --> 00:13:33.235 Now what have we got here? This is the second term in the 00:13:33.235 --> 00:13:39.773 series. But we've only got 1 two there, so if you like we've got 00:13:39.773 --> 00:13:43.141 1 + 2 - 1 times by two. 00:13:44.060 --> 00:13:48.776 One plus now, what's multiplying the two here? Well, this is the 00:13:48.776 --> 00:13:50.741 third term in the series. 00:13:51.650 --> 00:13:59.306 So we've got a 2 here, so we're multiplying by 3 - 00:13:59.306 --> 00:14:05.199 1. Here this is term #4 and we're 00:14:05.199 --> 00:14:10.932 multiplying by three, so that's 4 - 1 times 00:14:10.932 --> 00:14:17.302 by two. And here this is term #5, so we've 00:14:17.302 --> 00:14:23.035 got 1 + 5 - 1 times by two. 00:14:24.430 --> 00:14:28.670 Now, if we think about what's happening here. 00:14:31.120 --> 00:14:33.268 We're starting with A. 00:14:34.670 --> 00:14:37.610 And then on to the A. We're 00:14:37.610 --> 00:14:44.940 adding D. Then we're adding on another day, so that's a plus 00:14:44.940 --> 00:14:51.828 2D, and then we're adding on another D. So that's a plus 00:14:51.828 --> 00:14:57.214 3D. The question is, if we've got N terms in our sequence, 00:14:57.214 --> 00:15:02.450 then what's the last term? But if we look, we can see that the 00:15:02.450 --> 00:15:04.320 first term was just a. 00:15:04.880 --> 00:15:12.244 The second term was a plus, one D. The third term was a plus 00:15:12.244 --> 00:15:19.608 2D. The fourth term was a plus 3D, so the end term must be 00:15:19.608 --> 00:15:22.238 a plus N minus one. 00:15:22.750 --> 00:15:23.230 Gay. 00:15:25.200 --> 00:15:32.150 Now, this last term of our sequence, we often label 00:15:32.150 --> 00:15:35.625 L and call it the 00:15:35.625 --> 00:15:37.015 last term. 00:15:37.070 --> 00:15:39.650 Or 00:15:40.420 --> 00:15:42.530 The end. 00:15:43.140 --> 00:15:46.750 Turn. To be more mathematical 00:15:46.750 --> 00:15:51.368 about it. And one of the things that we'd like to be able to do 00:15:51.368 --> 00:15:54.566 with a sequence of numbers like this is get to a series. In 00:15:54.566 --> 00:15:58.256 other words, to be able to add them up. So let's have a look at 00:15:58.256 --> 00:16:05.693 that. So SN the some of these end terms is A plus A+B plus 00:16:05.693 --> 00:16:12.847 A plus 2B plus. But I want just to stop there and what I 00:16:12.847 --> 00:16:19.490 want to do is I want to start at the end. This end 00:16:19.490 --> 00:16:24.089 now now the last one will be plus L. 00:16:25.150 --> 00:16:29.518 So what will be the next one back when we generate each term 00:16:29.518 --> 00:16:35.230 by adding on D. So we added on D to this one to get L. So this 00:16:35.230 --> 00:16:37.582 one's got to be L minus D. 00:16:38.920 --> 00:16:45.750 And the one before that one similarly will be L 00:16:45.750 --> 00:16:51.520 minus 2D. On the rest of the terms will be in between. 00:16:52.640 --> 00:16:54.999 Now I'm going to use a trick. 00:16:55.540 --> 00:16:58.850 Mathematicians often use. I'm going to write this down the 00:16:58.850 --> 00:17:02.072 other way around. So I have L 00:17:02.072 --> 00:17:05.870 there. Plus L minus 00:17:05.870 --> 00:17:12.433 D. Plus L minus 2D plus plus. Now what will I have? 00:17:12.433 --> 00:17:17.869 Well, writing this down either way around, I'll Have A at the 00:17:17.869 --> 00:17:21.255 end. Then I'll have this next 00:17:21.255 --> 00:17:23.730 term a. Plus D. 00:17:24.230 --> 00:17:28.973 And I'll have this next term, A plus 2D. 00:17:31.490 --> 00:17:36.319 Now I'm going to add these two together. Let's look what 00:17:36.319 --> 00:17:41.587 happens if I add SN&SN together. I've just got two of them. 00:17:42.970 --> 00:17:49.548 By ad A&L together I get a plus L let me just group 00:17:49.548 --> 00:17:50.560 those together. 00:17:51.820 --> 00:17:59.004 Now I've got a plus D&L Minus D, so if I add them together I have 00:17:59.004 --> 00:18:06.188 a plus L Plus D minus D, so all I've got left is A plus L. 00:18:07.290 --> 00:18:14.192 But the same thing is going to happen here. I have a plus L 00:18:14.192 --> 00:18:19.122 Plus 2D Takeaway 2D, so again just a plus L. 00:18:20.030 --> 00:18:25.070 When we get down To this end, it's still the same thing 00:18:25.070 --> 00:18:30.110 happening. I've A plus L takeaway 2D add onto D so again 00:18:30.110 --> 00:18:35.570 the DS have disappeared. If you like and I've got L plus A. 00:18:36.480 --> 00:18:43.884 Plus a plusle takeaway D add on DLA and right at the 00:18:43.884 --> 00:18:46.969 end. L plus a again. 00:18:48.640 --> 00:18:53.333 Well, how many of these have I got? But I've got N terms. 00:18:54.130 --> 00:19:01.732 In each of these lines of sums, so I must still have end terms 00:19:01.732 --> 00:19:07.705 here, and so this must be an times a plus L. 00:19:08.390 --> 00:19:15.530 And so if we now divide both sides by two, we have. 00:19:15.530 --> 00:19:22.670 SN is 1/2 of N times by a plus L and that 00:19:22.670 --> 00:19:28.620 gives us our some of the terms of an arithmetic 00:19:28.620 --> 00:19:35.409 progression. Let's just write down again the two results that 00:19:35.409 --> 00:19:42.741 we've got. We've got L the end term, or the final term 00:19:42.741 --> 00:19:50.073 is equal to a plus N minus one times by D and 00:19:50.073 --> 00:19:53.739 we've got the SN is 1/2. 00:19:54.320 --> 00:20:01.380 Times by N number of terms times by a plus 00:20:01.380 --> 00:20:02.086 L. 00:20:03.290 --> 00:20:09.062 Now, one thing we can do is take this expression for L and 00:20:09.062 --> 00:20:10.838 substitute it into here. 00:20:11.770 --> 00:20:19.570 Replacing this al, so let's do that. SN is equal to 1/2. 00:20:20.180 --> 00:20:27.716 Times by N number of terms times by a plus and instead 00:20:27.716 --> 00:20:35.252 of L will write this a plus N minus one times by 00:20:35.252 --> 00:20:39.558 D. April say gives us two way. 00:20:40.600 --> 00:20:47.640 So the sum of the end terms is 1/2 an 00:20:47.640 --> 00:20:51.160 2A plus N minus 1D. 00:20:51.930 --> 00:20:53.958 Close the bracket. 00:20:55.730 --> 00:21:02.064 And these. That I'm underlining are the three 00:21:02.064 --> 00:21:05.838 important things about an arithmetic progression. 00:21:07.280 --> 00:21:11.070 If A is the first 00:21:11.070 --> 00:21:16.826 term. And D is the common difference. 00:21:17.750 --> 00:21:23.102 And N is the 00:21:23.102 --> 00:21:27.116 number of terms. 00:21:28.370 --> 00:21:33.545 In our arithmetic progression, then, this expression gives us 00:21:33.545 --> 00:21:39.870 the NTH or the last term. This expression gives us the 00:21:39.870 --> 00:21:46.195 some of those N terms, and this expression gives us also 00:21:46.195 --> 00:21:49.645 the sum of the end terms. 00:21:50.560 --> 00:21:54.520 One of the things that you also need to understand is that 00:21:54.520 --> 00:21:58.150 sometimes we like to shorten the language as well as using 00:21:58.150 --> 00:22:02.861 algebra. So that rather than keep saying arithmetic 00:22:02.861 --> 00:22:07.838 progression, we often refer to these as a peas. 00:22:09.130 --> 00:22:11.278 Now we've got some facts, some 00:22:11.278 --> 00:22:16.526 information there. So let's have a look at trying to see if we 00:22:16.526 --> 00:22:18.662 can use them to solve some 00:22:18.662 --> 00:22:25.800 questions. So let's have a look at this 00:22:25.800 --> 00:22:31.750 sequence of numbers again, which we've identified. 00:22:33.460 --> 00:22:37.016 And let's ask ourselves what's the sum? 00:22:38.150 --> 00:22:41.100 Of. The first 00:22:41.780 --> 00:22:48.278 50 terms So we could start to try and add 00:22:48.278 --> 00:22:54.374 them up. 1 + 3 is four and four and five is 9, and nine and 00:22:54.374 --> 00:22:59.708 Seven is 16 and 16 and 9025, and then the next get or getting 00:22:59.708 --> 00:23:03.899 rather complicated. But we can write down some facts about this 00:23:03.899 --> 00:23:08.471 straight away. We can write down that the first term is one. 00:23:09.070 --> 00:23:14.350 We can write down that the common difference Dean is 2 and 00:23:14.350 --> 00:23:20.070 we can write down the number of terms we're dealing with. An is 00:23:20.070 --> 00:23:27.041 50. We know we have a formula that says SN is 1/2 00:23:27.041 --> 00:23:29.696 times the number of terms. 00:23:30.710 --> 00:23:37.696 Times 2A plus N minus 1D. So instead of having to add this up 00:23:37.696 --> 00:23:43.684 as though it was a big arithmetic sum a big problem, we 00:23:43.684 --> 00:23:48.674 can simply substitute the numbers into the formula. So SNS 00:23:48.674 --> 00:23:52.666 50 in this case is equal to 1/2. 00:23:53.080 --> 00:23:55.288 Times by 50. 00:23:55.800 --> 00:24:02.940 Times by two A That's just two 2 * 1 plus N minus one 00:24:02.940 --> 00:24:06.510 and is 50, so N minus one 00:24:06.510 --> 00:24:10.210 is 49. Times by the common 00:24:10.210 --> 00:24:11.270 difference too. 00:24:12.260 --> 00:24:19.716 So. We can cancel a 2 into the 50 00:24:19.716 --> 00:24:23.230 that gives us 25 times by now. 00:24:23.770 --> 00:24:30.840 2 * 49 or 2 * 49 is 98 and two is 100, so 00:24:30.840 --> 00:24:37.405 we have 25 times by 100, so that's 2500. So what was going 00:24:37.405 --> 00:24:42.455 to be quite a lengthy and difficult calculation's come out 00:24:42.455 --> 00:24:48.702 quite quickly. Let's see if we can solve a more difficult 00:24:48.702 --> 00:24:53.000 problem. 1. 00:24:54.110 --> 00:24:58.840 Plus 3.5. +6. 00:25:00.050 --> 00:25:03.620 Plus 8.5. Plus 00:25:04.490 --> 00:25:08.270 Plus 101. 00:25:10.290 --> 00:25:11.310 Add this up. 00:25:12.490 --> 00:25:19.552 Well. Can we identify what kind of a series this is? We can 00:25:19.552 --> 00:25:25.155 see quite clearly that one to 3.5 while that's a gap of 2.5 00:25:25.155 --> 00:25:32.051 and then a gap of 2.5 to 6. So what we've got here is in fact 00:25:32.051 --> 00:25:36.792 an arithmetic progression, and we can see here. We've got 100 00:25:36.792 --> 00:25:42.826 and one at the end. Our last term is 101 and the first term 00:25:42.826 --> 00:25:45.843 is one. Now we know a formula. 00:25:45.890 --> 00:25:49.390 For the last term L. 00:25:50.050 --> 00:25:57.133 Equals A plus N minus one times by D. 00:25:58.350 --> 00:26:05.126 Might just have a look at what we know in this formula. What we 00:26:05.126 --> 00:26:07.062 know L it's 101. 00:26:07.070 --> 00:26:12.227 We know a It's the first term, it's one. 00:26:13.280 --> 00:26:19.259 Plus Well, we have no idea what any is. We don't know how many 00:26:19.259 --> 00:26:24.061 terms we've got, so that's N minus one times by D and we know 00:26:24.061 --> 00:26:25.776 what that is, that's 2.5. 00:26:26.510 --> 00:26:31.790 Well, this is nothing more than an equation for an, so let's 00:26:31.790 --> 00:26:37.510 begin by taking one from each side. That gives us 100 equals N 00:26:37.510 --> 00:26:43.230 minus one times by 2.5. And now I'm going to divide both sides 00:26:43.230 --> 00:26:49.830 by 2.5 and that will give me 40 equals N minus one, and now I'll 00:26:49.830 --> 00:26:57.310 add 1 to both sides and so 41 is equal to end, so I know how many 00:26:57.310 --> 00:27:04.198 terms that. Are in this series, So what I can do now is I 00:27:04.198 --> 00:27:10.186 can add it up because the sum of N terms is 1/2. 00:27:10.860 --> 00:27:13.872 NA plus 00:27:13.872 --> 00:27:20.800 L. And I now know all these terms 00:27:20.800 --> 00:27:24.300 here have 1/2 * 41 00:27:24.300 --> 00:27:27.485 * 1. Plus 00:27:27.485 --> 00:27:34.790 101. Let me just turn the page over and write this 00:27:34.790 --> 00:27:36.458 some down again. 00:27:37.120 --> 00:27:43.564 SN is equal to 1/2 * 00:27:43.564 --> 00:27:48.934 41 * 1 + 101. 00:27:50.040 --> 00:27:57.432 So we have 1/2 times by 41 times by 102 and we can cancel 00:27:57.432 --> 00:28:04.824 it to there to give US 41 times by 51. And to do that 00:28:04.824 --> 00:28:10.632 I'd want to get out my Calculator, but we'll leave it 00:28:10.632 --> 00:28:12.744 there to be finished. 00:28:13.270 --> 00:28:16.858 So that's one kind of problem. 00:28:17.800 --> 00:28:21.877 Let's have a look at another kind of problem. 00:28:22.430 --> 00:28:28.214 Let's say we've got an arithmetic progression whose 00:28:28.214 --> 00:28:31.106 first term is 3. 00:28:32.170 --> 00:28:35.530 And the sum. 00:28:36.200 --> 00:28:37.570 Of. 00:28:39.210 --> 00:28:42.970 The first 8. 00:28:44.740 --> 00:28:45.630 Terms. 00:28:47.020 --> 00:28:53.972 Is twice. The sum 00:28:53.972 --> 00:28:59.736 of the first 5 00:28:59.736 --> 00:29:01.177 terms. 00:29:02.570 --> 00:29:04.820 And that seems really quite 00:29:04.820 --> 00:29:09.228 complicated. But it needn't be, but remember this is the 00:29:09.228 --> 00:29:10.374 same arithmetic progression. 00:29:12.250 --> 00:29:18.018 So let's have a think what this is telling us A is equal to 00:29:18.018 --> 00:29:23.374 three and the sum of the first 8 terms. Well, to begin with, 00:29:23.374 --> 00:29:28.318 let's write down what the sum of the first 8 terms is. 00:29:28.870 --> 00:29:31.818 Well, it's a half. 00:29:32.440 --> 00:29:39.088 Times N Times 2A plus and 00:29:39.088 --> 00:29:41.304 minus 1D. 00:29:42.470 --> 00:29:45.848 And N is equal to 8. 00:29:46.930 --> 00:29:48.750 So we've got a half. 00:29:49.980 --> 00:29:57.630 Times 8. 2A plus N minus one is 00:29:57.630 --> 00:30:05.484 7D. So S 8 is equal to half of 00:30:05.484 --> 00:30:11.828 eight is 4 * 2 A Plus 7D. 00:30:12.970 --> 00:30:20.434 But we also know that a is equal to three, so we 00:30:20.434 --> 00:30:27.898 can put that in there as well. That's 4 * 6 because 00:30:27.898 --> 00:30:31.630 a is 3 + 7 D. 00:30:32.340 --> 00:30:39.812 Next one, the sum of the first 5 00:30:39.812 --> 00:30:46.834 terms. Let me just write down some of the first 00:30:46.834 --> 00:30:48.862 8 terms were. 00:30:49.150 --> 00:30:55.554 4. Times 6 minus 00:30:55.554 --> 00:31:01.146 plus 7D first 5 00:31:01.146 --> 00:31:08.706 terms. Half times the number of terms. That's 5 * 2 A plus 00:31:08.706 --> 00:31:15.888 N minus one times by D will. That must be 4 because any is 00:31:15.888 --> 00:31:17.940 5 times by D. 00:31:18.900 --> 00:31:25.478 So much is 5 over 2 and let's remember that a is equal 00:31:25.478 --> 00:31:32.562 to three, so that 6 + 4 D. So I've got S 8 and 00:31:32.562 --> 00:31:39.140 I've got S5 and the question said that S8 was equal to twice 00:31:39.140 --> 00:31:42.970 as five. So I can write this 00:31:42.970 --> 00:31:48.635 for S8. Is equal to 00:31:48.635 --> 00:31:56.030 twice. This which is S five 2 * 5 over two 00:31:56.030 --> 00:32:02.306 6 + 4 D and what seemed a very difficult question as 00:32:02.306 --> 00:32:08.059 reduced itself to an ordinary linear equation in terms of D. 00:32:08.059 --> 00:32:14.335 So we can do some cancelling there and we can multiply out 00:32:14.335 --> 00:32:21.657 the brackets for six is a 24 + 28, D is equal to 56R. 00:32:21.690 --> 00:32:24.974 30 + 5 fours 00:32:24.974 --> 00:32:31.870 are 20D. I can take 20D from each side that gives me 00:32:31.870 --> 00:32:33.256 8 D there. 00:32:33.830 --> 00:32:40.998 And I can take 24 from each side, giving me six there. So D 00:32:40.998 --> 00:32:42.534 is equal to. 00:32:43.650 --> 00:32:49.994 Dividing both sides by 8, six over 8 or 3/4 so I know 00:32:49.994 --> 00:32:54.874 everything now that I could possibly want to know about 00:32:54.874 --> 00:32:56.338 this arithmetic progression. 00:32:57.940 --> 00:33:04.142 Now let's go on and have a look at our second type of special 00:33:04.142 --> 00:33:05.471 sequence, a geometric 00:33:05.471 --> 00:33:11.398 progression. So. Take these 00:33:11.398 --> 00:33:14.666 two six 00:33:14.666 --> 00:33:20.680 1854. Let's have a look at how this sequence of numbers 00:33:20.680 --> 00:33:23.207 is growing. We have two. Then we 00:33:23.207 --> 00:33:31.060 have 6. And then we have 18. Well 326 and three sixes 00:33:31.060 --> 00:33:38.896 are 18 and three eighteens are 54. So this sequence is growing 00:33:38.896 --> 00:33:45.426 by multiplying by three each time. What about this sequence 00:33:45.426 --> 00:33:48.436 one? Minus 00:33:48.436 --> 00:33:52.215 2 four. Minus 00:33:52.215 --> 00:33:57.040 8. What's happening here? We can see the signs are alternating, 00:33:57.040 --> 00:33:58.996 but let's just look at the 00:33:58.996 --> 00:34:05.067 numbers. 1 * 2 would be two 2 * 2 would be four. 2 * 4 would be 00:34:05.067 --> 00:34:10.890 8. But if we made that minus two, then one times minus two 00:34:10.890 --> 00:34:17.945 would be minus 2 - 2 times minus two would be plus 4 + 4 times by 00:34:17.945 --> 00:34:23.340 minus two would be minus 8, so this sequence to be generated is 00:34:23.340 --> 00:34:27.905 being multiplied by minus two. Each term is multiplied by minus 00:34:27.905 --> 00:34:30.395 two to give the next term. 00:34:31.210 --> 00:34:36.840 These are examples of geometric progressions, or if you like, 00:34:36.840 --> 00:34:42.830 GPS. Let's try and write one down in general using some 00:34:42.830 --> 00:34:48.810 algebra. So like the AP, we take A to be the first term. 00:34:49.640 --> 00:34:54.249 Now we need something like D. The common difference, but what 00:34:54.249 --> 00:35:00.115 we use is the letter R and we call it the common ratio, and 00:35:00.115 --> 00:35:05.143 that's the number that does the multiplying of each term to give 00:35:05.143 --> 00:35:06.400 the next term. 00:35:07.090 --> 00:35:14.395 So 3 times by two gives us 6, so that's the R. In this case 00:35:14.395 --> 00:35:18.291 the three. So we do a Times by 00:35:18.291 --> 00:35:24.999 R. And then we multiply by, in this case by three again 3 times 00:35:24.999 --> 00:35:29.955 by 6 gives 18, so we multiply by R again, AR squared. 00:35:30.670 --> 00:35:38.122 And then we multiply by three again to give us the 54. 00:35:38.122 --> 00:35:41.848 So by our again AR cubed. 00:35:42.680 --> 00:35:49.638 And what's our end term in this case? While A is the first term 00:35:49.638 --> 00:35:56.596 8 times by R, is the second term 8 times by R-squared is the 00:35:56.596 --> 00:36:03.554 third term 8 times by R cubed? Is the fourth term, so it's a 00:36:03.554 --> 00:36:10.015 times by R to the N minus one. Because this power there's a 00:36:10.015 --> 00:36:12.997 one. There is always one less. 00:36:13.000 --> 00:36:17.268 And the number of the term, then its position in the 00:36:17.268 --> 00:36:22.700 sequence. And this is the end term, so it's a Times my R to 00:36:22.700 --> 00:36:24.252 the N minus one. 00:36:25.470 --> 00:36:31.766 What about adding up a geometric progression? Let's 00:36:31.766 --> 00:36:39.636 write that down. SN is equal to a plus R 00:36:39.636 --> 00:36:41.997 Plus R-squared Plus. 00:36:42.580 --> 00:36:50.070 Plus AR to the N minus one, and that's the sum of N terms. 00:36:50.810 --> 00:36:56.342 Going to use another trick similar but not the same to what 00:36:56.342 --> 00:37:00.491 we did with arithmetic progressions. What I'm going to 00:37:00.491 --> 00:37:05.562 do is I'm going to multiply everything by the common ratio. 00:37:06.590 --> 00:37:11.894 So I've multiplied SN by are going to multiply this one by R, 00:37:11.894 --> 00:37:17.198 but I'm not going to write the answer there. I'm going to write 00:37:17.198 --> 00:37:23.726 it here so I've a Times by R and I've written it there plus now I 00:37:23.726 --> 00:37:29.030 multiply this one by R and that would give me a R-squared. I'm 00:37:29.030 --> 00:37:31.070 going to write it there. 00:37:31.630 --> 00:37:36.089 So that term is being multiplied by R and it's gone to their 00:37:36.089 --> 00:37:40.205 that's being multiplied by R and it's gone to their. This one 00:37:40.205 --> 00:37:45.350 will be multiplied by R and it will be a R cubed and it will 00:37:45.350 --> 00:37:46.722 have gone to their. 00:37:47.360 --> 00:37:52.427 Plus etc plus, and we think about what's happening. 00:37:53.350 --> 00:37:58.250 That term will come to here and it will look just like that one. 00:37:59.010 --> 00:38:03.582 Plus and then we need to multiply this by R, and that's 00:38:03.582 --> 00:38:07.011 another. Are that we're multiplying by, so that means 00:38:07.011 --> 00:38:09.297 that becomes AR to the N. 00:38:10.230 --> 00:38:16.548 Now look at why I've lined these up AR, AR, AR squared. Our 00:38:16.548 --> 00:38:19.464 squared, al, cubed, cubed and so 00:38:19.464 --> 00:38:25.423 on. So let's take these two lines of algebra away from each 00:38:25.423 --> 00:38:31.625 other, so I'll have SN minus R times by SN is equal to. Now 00:38:31.625 --> 00:38:38.270 have nothing here to take away from a, so the a stays as it is. 00:38:38.270 --> 00:38:42.700 Then I've AR takeaway are, well, that's nothing. A R-squared 00:38:42.700 --> 00:38:46.244 takeaway R-squared? That's nothing again, same there. And 00:38:46.244 --> 00:38:50.231 so on and so on. AR to the N 00:38:50.231 --> 00:38:55.054 minus one. Take away a art. The end minus one nothing and then 00:38:55.054 --> 00:38:57.532 at the end I have nothing there 00:38:57.532 --> 00:39:02.018 take away. AR to the N. 00:39:03.240 --> 00:39:07.049 Now I need to look closely at both sides of what I've got 00:39:07.049 --> 00:39:10.858 written down, and I'm going to turn this over and write it down 00:39:10.858 --> 00:39:18.516 again. So we've SN minus RSN is equal to A. 00:39:19.020 --> 00:39:22.240 Minus AR to the N. 00:39:22.930 --> 00:39:28.429 Now here I've got a common factor SN the some of the end 00:39:28.429 --> 00:39:34.351 terms when I take that out, I've won their minus R of them there, 00:39:34.351 --> 00:39:41.542 so I get SN times by one minus R is equal 2 and here I've got a 00:39:41.542 --> 00:39:48.310 common Factor A and I can take a out giving me one minus R to the 00:39:48.310 --> 00:39:53.809 N. Remember it was the sum of N terms that I wanted so. 00:39:53.860 --> 00:40:00.466 SN is equal to a Times 1 minus R to the N and to get the SN on 00:40:00.466 --> 00:40:05.604 its own, I've had to divide by one minus R, so I must divide 00:40:05.604 --> 00:40:07.439 this by one minus R. 00:40:09.370 --> 00:40:15.844 And that's my formula for the sum of N terms of a geometric 00:40:15.844 --> 00:40:19.828 progression. And let's just remind ourselves what the 00:40:19.828 --> 00:40:24.808 symbols are N is equal to the number of terms. 00:40:24.820 --> 00:40:30.847 A is the first term of our 00:40:30.847 --> 00:40:34.291 geometric progression and are 00:40:34.291 --> 00:40:40.318 we said was called the common ratio. 00:40:41.420 --> 00:40:48.011 OK, and let's just remember the NTH term in the sequence was AR 00:40:48.011 --> 00:40:55.109 to the N minus one. So those are our fax so far about GPS 00:40:55.109 --> 00:41:00.686 or geometric progressions. Let's see if we can use these facts 00:41:00.686 --> 00:41:07.277 in order to be able to help us solve some problems and do 00:41:07.277 --> 00:41:15.027 some questions. So first of all, let's take this 2 + 6 + 00:41:15.027 --> 00:41:22.268 18 + 54 plus. Let's say there are six terms. What's the answer 00:41:22.268 --> 00:41:25.610 when it comes to adding those 00:41:25.610 --> 00:41:32.550 up? Well, we know that a is equal to two. We know that 00:41:32.550 --> 00:41:39.825 our is equal to three and we know that N is equal to six. So 00:41:39.825 --> 00:41:47.585 to solve that, all we need to do is write down that the sum of N 00:41:47.585 --> 00:41:54.860 terms is a Times 1 minus R to the N all over 1 minus R. 00:41:54.860 --> 00:41:57.285 Substitute our numbers in two 00:41:57.285 --> 00:42:03.904 times. 1 - 3 to the power 6. 00:42:04.520 --> 00:42:07.991 Over 1 - 00:42:07.991 --> 00:42:14.612 3. So this is 2 * 1 - 3 to the power six over minus 00:42:14.612 --> 00:42:20.044 two, and we can cancel a minus two with the two that we leave 00:42:20.044 --> 00:42:24.700 as with a minus one there and one there if I multiply 00:42:24.700 --> 00:42:29.744 throughout by the minus one, I'll have minus 1 * 1 is minus 00:42:29.744 --> 00:42:35.952 one and minus one times minus 3 to the six is 3 to the 6th, so 00:42:35.952 --> 00:42:39.444 the sum of N terms is 3 to the 00:42:39.444 --> 00:42:45.430 power 6. Minus one and with a Calculator we could workout what 00:42:45.430 --> 00:42:49.056 3 to the power 6 - 1 00:42:49.056 --> 00:42:52.256 was. Let's take 00:42:52.256 --> 00:42:59.440 another. Question to do with summing the terms of a geometric 00:42:59.440 --> 00:43:05.892 progression. What's the sum of that? Let's say for five 00:43:05.892 --> 00:43:11.772 terms. While we can begin by identifying the first term, 00:43:11.772 --> 00:43:15.888 that's eight, and what's the common ratio? 00:43:17.100 --> 00:43:23.100 Well, to go from 8 to 4 as a number we would have it, but 00:43:23.100 --> 00:43:27.900 there's a minus sign in there. So that suggests that the common 00:43:27.900 --> 00:43:32.700 ratio is minus 1/2. Let's just check it minus four times. By 00:43:32.700 --> 00:43:39.100 minus 1/2 is plus 2 + 2 times Y minus 1/2 is minus one, and we 00:43:39.100 --> 00:43:41.900 said five terms, so Ann is equal 00:43:41.900 --> 00:43:49.494 to 5. So we can write down our formula. SN is equal to 00:43:49.494 --> 00:43:57.390 a Times 1 minus R to the power N all over 1 minus R. 00:43:57.980 --> 00:44:01.120 And so A is 8. 00:44:02.560 --> 00:44:10.220 1 minus and this is minus 1/2 to the power 00:44:10.220 --> 00:44:16.460 5. All over 1 minus minus 1/2. You can see these 00:44:16.460 --> 00:44:19.328 questions get quite complicated with the 00:44:19.328 --> 00:44:24.108 arithmetic, so you have to be very careful and you 00:44:24.108 --> 00:44:27.932 have to have a good knowledge of fractions. 00:44:29.320 --> 00:44:37.112 This is 8 * 1. Now let's have a look at minus 1/2 to the power 00:44:37.112 --> 00:44:42.360 5. Well, I'm multiplying the minus sign by itself five times, 00:44:42.360 --> 00:44:47.136 which would give me a negative number, and I've got a minus 00:44:47.136 --> 00:44:51.912 sign there outside the bracket. That's going to mean I've got 6 00:44:51.912 --> 00:44:57.086 minus signs together. Makes it plus. So now I can look at the 00:44:57.086 --> 00:44:59.076 half to the power 5. 00:44:59.670 --> 00:45:02.574 Well, that's going to be one 00:45:02.574 --> 00:45:09.270 over. 248-1632 to to the power 00:45:09.270 --> 00:45:12.696 five is 32. 00:45:13.400 --> 00:45:20.504 All over 1 minus minus 1/2. That's 1 + 1/2. Let's write 00:45:20.504 --> 00:45:23.464 that as three over 2. 00:45:24.540 --> 00:45:27.170 So this is equal to. 00:45:28.160 --> 00:45:30.700 Now I've got 8. 00:45:31.450 --> 00:45:37.764 Times by one plus, one over 32, and I'm dividing by three over 2 00:45:37.764 --> 00:45:42.725 to divide by a fraction. We invert the fraction that's two 00:45:42.725 --> 00:45:49.039 over 3 and we multiply by and we just turn the page to finish 00:45:49.039 --> 00:45:50.392 this one off. 00:45:51.110 --> 00:45:58.840 So we have SN is equal to 8 * 1 00:45:58.840 --> 00:46:06.570 + 1 over 32 times by 2/3 is equal to 00:46:06.570 --> 00:46:14.300 8 times by now one and 132nd. Well, there are 00:46:14.300 --> 00:46:22.030 3230 seconds in one, so altogether there I've got 33. 00:46:22.030 --> 00:46:25.264 30 seconds times 00:46:25.264 --> 00:46:30.926 by 2/3. And we can do some canceling threes 00:46:30.926 --> 00:46:35.582 into 30. Three will go 11 and threes into three. There goes 00:46:35.582 --> 00:46:42.536 one. Twos into two goes one and tools into 32, goes 16 and 18 00:46:42.536 --> 00:46:48.864 two eight goes one and eight into 16 goes 2. So we 1 * 00:46:48.864 --> 00:46:55.192 11 * 1 that's just 11 over 2 because we've 2 times by one 00:46:55.192 --> 00:47:01.068 there. So we love Nova two or we prefer five and a half. 00:47:01.840 --> 00:47:07.216 So that we've got the some of those five terms of that 00:47:07.216 --> 00:47:12.144 particular GP. Five and a half, 11 over 2 or 5.5. 00:47:12.680 --> 00:47:19.775 But here's a different question. What if we've got the sequence 00:47:19.775 --> 00:47:27.512 248? 128 how many terms are we got? How many bits do we 00:47:27.512 --> 00:47:33.944 need to get from 2 up to 128? Well, let's begin by 00:47:33.944 --> 00:47:36.088 identifying the first term 00:47:36.088 --> 00:47:39.610 that's two. This is. 00:47:40.820 --> 00:47:45.792 A geometric progression because we multiply by two to get each 00:47:45.792 --> 00:47:52.120 term. So the common ratio are is 2 and what we don't know is 00:47:52.120 --> 00:47:59.168 what's N. So let's have a look. This is the last term and we 00:47:59.168 --> 00:48:05.524 know our expression for the last term. 128 is equal to AR to the 00:48:05.524 --> 00:48:06.886 N minus one. 00:48:07.700 --> 00:48:14.427 So let's substituting some of our information. 00:48:14.450 --> 00:48:22.310 A is 2 times by two 4R to the N minus one. 00:48:23.010 --> 00:48:29.020 Well, we can divide both sides by this two here, 00:48:29.020 --> 00:48:31.424 which will give us. 00:48:32.570 --> 00:48:38.240 64 is equal to two to the N minus one. 00:48:38.870 --> 00:48:45.170 I think about that it's 248 sixteen 3264 so I had to 00:48:45.170 --> 00:48:51.995 multiply 2 by itself six times in order to get 64, so 2 00:48:51.995 --> 00:48:59.345 to the power 6, which is 64 is equal to 2 to the power 00:48:59.345 --> 00:49:06.695 N minus one, so six is equal to N minus one, and so N 00:49:06.695 --> 00:49:09.845 is equal to 7, adding one. 00:49:09.860 --> 00:49:14.960 To each side. In other words, there were Seven terms in our. 00:49:15.520 --> 00:49:22.122 Geometric progression. Type of question that's often given 00:49:22.122 --> 00:49:26.698 for geometric progressions is given a geometric progression. 00:49:26.698 --> 00:49:32.990 How many terms do you need to add together before you 00:49:32.990 --> 00:49:38.710 exceed a certain limit? So, for instance, here's a geometric. 00:49:38.820 --> 00:49:44.832 Progression. How many times of this geometric progression do we 00:49:44.832 --> 00:49:50.684 need to act together in order to be sure that the some of them 00:49:50.684 --> 00:49:52.356 will get over 20? 00:49:53.040 --> 00:49:58.932 Well, first of all, let's try and identify this as a geometric 00:49:58.932 --> 00:50:04.333 progression. The first term is on and it looks like what's 00:50:04.333 --> 00:50:09.243 doing the multiplying. The common ratio is 1.1. Let's just 00:50:09.243 --> 00:50:10.716 check that here. 00:50:11.290 --> 00:50:16.919 1.1 times by one point, one well. That's kind of like 11 * 00:50:16.919 --> 00:50:18.218 11 is 121. 00:50:18.730 --> 00:50:23.086 With two numbers after the decimal point in one point 1 * 00:50:23.086 --> 00:50:27.442 1.1 and with two numbers after the decimal point there. So yes, 00:50:27.442 --> 00:50:29.257 this is a geometric progression. 00:50:30.050 --> 00:50:36.719 So let's write down our formula for N terms sum of N terms 00:50:36.719 --> 00:50:42.362 is equal to a Times 1 minus R to the N. 00:50:42.970 --> 00:50:50.320 All over 1 minus R. We want to know what value of N is just 00:50:50.320 --> 00:50:53.260 going to take us over 20. 00:50:53.880 --> 00:51:00.090 So let's substituting some numbers. This is one for 00:51:00.090 --> 00:51:03.540 a 1 - 1.1 to 00:51:03.540 --> 00:51:10.816 the N. All over 1 - 1.1 that 00:51:10.816 --> 00:51:15.952 has to be greater than 20. 00:51:17.210 --> 00:51:22.886 So one times by that isn't going to affect what's in the 00:51:22.886 --> 00:51:29.981 brackets. That would be 1 - 1.1 to the N all over 1 - 1.1 00:51:29.981 --> 00:51:35.184 is minus nought. .1 that has to be greater than 20. 00:51:36.510 --> 00:51:42.256 Now if I use the minus sign wisely. In other words, If I 00:51:42.256 --> 00:51:44.024 divide if you like. 00:51:45.280 --> 00:51:48.724 Minus note .1 into there as 00:51:48.724 --> 00:51:52.110 a. Division, then I'll have. 00:51:52.690 --> 00:51:58.878 The minus sign will make that a minus and make that a plus, so 00:51:58.878 --> 00:52:05.066 I'll have one point 1 to the N minus one and divided by North 00:52:05.066 --> 00:52:10.370 Point one is exactly the same as multiplying by 10. That means 00:52:10.370 --> 00:52:12.580 I've got a 10 here. 00:52:13.310 --> 00:52:16.467 That I can divide both sides by. 00:52:17.620 --> 00:52:23.613 So let's just write this down again 1.1 to the N minus one 00:52:23.613 --> 00:52:29.606 times by 10 has to be greater than 20. So let's divide both 00:52:29.606 --> 00:52:36.521 sides by 10, one point 1 to the N minus one has to be greater 00:52:36.521 --> 00:52:43.897 than two and will add the one to both sides 1.1 to the end has to 00:52:43.897 --> 00:52:45.741 be greater than three. 00:52:46.550 --> 00:52:52.010 Problem how do we find N? One of the ways of solving equations 00:52:52.010 --> 00:52:57.470 like this is to take logarithms of both sides, so I'm going to 00:52:57.470 --> 00:53:02.510 take natural logarithms of both sides. I'm going to do it to 00:53:02.510 --> 00:53:07.130 this site first. That's the natural logarithm of 3 N about 00:53:07.130 --> 00:53:12.590 this side. When you're taking a log of a number that's raised to 00:53:12.590 --> 00:53:17.210 the power, that's the equivalent of multiplying the log of that 00:53:17.210 --> 00:53:23.191 number. By the power that's N times the log of 1.1. Well 00:53:23.191 --> 00:53:29.704 now this is just an equation for N because N has got to 00:53:29.704 --> 00:53:36.217 be greater than the log of 3 divided by the log of 1.1 00:53:36.217 --> 00:53:37.720 because after all. 00:53:39.200 --> 00:53:44.645 Log of three is just a number and log of 1.1 is just a number 00:53:44.645 --> 00:53:48.275 and this is the sort of calculation that really does 00:53:48.275 --> 00:53:50.816 have to be done on a Calculator. 00:53:51.800 --> 00:53:57.146 So if we take our Calculator and we turn it on. 00:53:58.360 --> 00:54:02.370 And we do the calculation. The natural log of three. 00:54:03.930 --> 00:54:09.678 Divided by the natural log of 1.1, we ask our Calculator to 00:54:09.678 --> 00:54:16.384 calculate that for us. It tells us that it's 11.5 to 6 and some 00:54:16.384 --> 00:54:20.695 more decimal places. We're not really worried about these 00:54:20.695 --> 00:54:26.922 decimal places. An is a whole number and it has to be greater 00:54:26.922 --> 00:54:33.628 than 11 and some bits, so N has got to be 12 or more. 00:54:35.970 --> 00:54:43.458 That's one last twist to our geometric progression. 00:54:43.460 --> 00:54:46.900 Let's have a look at 00:54:46.900 --> 00:54:53.825 this one. What have we got got 00:54:53.825 --> 00:55:00.479 a geometric progression. First term a 00:55:00.479 --> 00:55:02.697 is one. 00:55:04.360 --> 00:55:10.910 Common ratio is 1/2 because we're multiplying by 1/2 each 00:55:10.910 --> 00:55:14.800 time. That write down 00:55:14.800 --> 00:55:21.456 some sums. S1, the sum of the first term is just. 00:55:22.060 --> 00:55:25.345 1. What's 00:55:25.345 --> 00:55:32.665 S2? That's the sum of the first 2 terms, so 00:55:32.665 --> 00:55:37.070 that's. Three over 2. 00:55:38.460 --> 00:55:43.420 What's the sum of the first three terms? That's one. 00:55:44.560 --> 00:55:48.106 Plus 1/2 + 00:55:48.106 --> 00:55:55.781 1/4. Add those up in terms of how many quarters 00:55:55.781 --> 00:55:59.555 have we got then that is 00:55:59.555 --> 00:56:06.695 7. Quarters As for the sum of 00:56:06.695 --> 00:56:08.505 the first. 00:56:08.760 --> 00:56:15.339 4. Terms. Add those up in terms of how 00:56:15.339 --> 00:56:21.037 many eighths if we got so we've got eight of them there. Four of 00:56:21.037 --> 00:56:26.328 them there. That's 12. Two of them there. That's 14 and one of 00:56:26.328 --> 00:56:28.363 them there. That's 15 eighths. 00:56:29.090 --> 00:56:31.978 Seems to be some sort of pattern here. 00:56:32.990 --> 00:56:36.374 Here we seem to be 1/2 short of 00:56:36.374 --> 00:56:42.690 two. Here we seem to 00:56:42.690 --> 00:56:49.410 be 1/4. Short of two here, we seem to be an eighth 00:56:49.410 --> 00:56:55.052 short of two and we look at the first one. Then we're clearly 1 00:56:55.052 --> 00:56:56.261 short of two. 00:56:56.930 --> 00:57:04.014 He's a powers of two. Let's have a look 2 - 2 to the 00:57:04.014 --> 00:57:09.580 power zero, 'cause 2 to the power zero is 1 two. 00:57:10.310 --> 00:57:11.510 Minus. 00:57:12.750 --> 00:57:20.660 2 to the power minus one 2 - 2 to the power minus two 00:57:20.660 --> 00:57:24.615 2 - 2 to the power minus 00:57:24.615 --> 00:57:29.414 three. But each of these is getting smaller. We're getting 00:57:29.414 --> 00:57:34.538 nearer and nearer to two. The next one we take away will be a 00:57:34.538 --> 00:57:40.028 16th, the one after that will be a 32nd and the next bit we take 00:57:40.028 --> 00:57:45.884 off 2 is going to be a 64th and then a 128 and then at one 00:57:45.884 --> 00:57:50.276 256th. So we're getting the bits were taking away from two are 00:57:50.276 --> 00:57:53.570 getting smaller and smaller and smaller until eventually we 00:57:53.570 --> 00:57:56.498 wouldn't be able to distinguish them from zero. 00:57:56.510 --> 00:58:01.592 And so if we could Add all of these up forever, a sum to 00:58:01.592 --> 00:58:06.674 Infinity, if you like the answer, or to be 2 or as near as 00:58:06.674 --> 00:58:12.482 we want to be to two. So let's see if we can have a look at 00:58:12.482 --> 00:58:13.934 that with some algebra. 00:58:14.760 --> 00:58:22.208 We know that the sum to end terms is equal to a Times 1 00:58:22.208 --> 00:58:27.528 minus R to the N all over 1 minus R. 00:58:28.050 --> 00:58:33.118 What we want to have a look at is this thing are because what 00:58:33.118 --> 00:58:34.566 was crucial about this? 00:58:35.960 --> 00:58:42.912 Geometric progression was at the common ratio was a half a 00:58:42.912 --> 00:58:45.440 number less than one. 00:58:45.450 --> 00:58:48.192 So let's have a look what 00:58:48.192 --> 00:58:55.928 happens. When all is bigger than one to R to the power N. 00:58:56.630 --> 00:59:01.778 We are is bigger than one and we keep multiplying it by itself. 00:59:02.280 --> 00:59:08.737 Grows, it grows very rapidly and really gets very big very 00:59:08.737 --> 00:59:15.781 quickly. Check it with two, 2, four, 816. It goes off til 00:59:15.781 --> 00:59:20.880 Infinity. And because it goes off to Infinity, it takes the 00:59:20.880 --> 00:59:22.875 sum with it as well. 00:59:24.350 --> 00:59:28.232 What about if our is equal 00:59:28.232 --> 00:59:34.380 to 1? Well, we can't really use this formula then because we 00:59:34.380 --> 00:59:39.024 would be dividing by zero. But if you think about it, are 00:59:39.024 --> 00:59:44.442 equals 1 means every term is the same. So if we start off with 00:59:44.442 --> 00:59:49.860 one every term is the same 1111 and you just add them all up. 00:59:49.860 --> 00:59:55.278 But again that means the sum is going to go off to Infinity if 00:59:55.278 --> 00:59:59.535 you take the number any number and add it to itself. 01:00:00.340 --> 01:00:04.504 An infinite number of times you're going to get a very, very 01:00:04.504 --> 01:00:12.083 big number. What happens if our is less than minus one? 01:00:12.083 --> 01:00:14.711 Something like minus 2? 01:00:15.450 --> 01:00:19.751 Well, what's going to happen then to R to the N? 01:00:20.920 --> 01:00:26.058 Well, it's going to be plus an. It's going to be minus as we 01:00:26.058 --> 01:00:31.196 multiply by this number such as minus two. So we have minus 2 + 01:00:31.196 --> 01:00:35.600 4 minus A. The thing to notice is it's getting bigger, it's 01:00:35.600 --> 01:00:40.371 getting bigger each time. So again are to the end is going to 01:00:40.371 --> 01:00:44.041 go off to Infinity. It's going to oscillate between plus 01:00:44.041 --> 01:00:48.445 Infinity and minus Infinity, but it's going to get very big and 01:00:48.445 --> 01:00:51.748 that means this sum is also going to get. 01:00:51.750 --> 01:00:52.490 Very big. 01:00:53.570 --> 01:01:00.086 What about our equals minus one? Well, if R equals minus one, 01:01:00.086 --> 01:01:05.516 let's think about a sequence like that. Well, a typical 01:01:05.516 --> 01:01:08.774 sequence might be 1 - 1. 01:01:09.490 --> 01:01:14.092 1 - 1 and we can see the problem. It depends where we 01:01:14.092 --> 01:01:19.402 stop. If I stop here the sum is 0 but if I put another one 01:01:19.402 --> 01:01:24.004 there, the sum is one. So we've got an infinite number of terms 01:01:24.004 --> 01:01:28.606 then. Well, it depends on money I've got us to what the answer 01:01:28.606 --> 01:01:33.562 is so there isn't a limit for SN. There isn't a thing that it 01:01:33.562 --> 01:01:35.686 can come to a definite number. 01:01:36.670 --> 01:01:40.963 Let's have a look. We've considered all possible values 01:01:40.963 --> 01:01:46.687 of our except those where are is between plus and minus one. 01:01:47.230 --> 01:01:50.270 Let's take our equals 1/2 as an example. 01:01:51.590 --> 01:01:54.446 Or half trans by half is 1/4. 01:01:55.470 --> 01:01:59.208 Reply by 1/2. Again that's an eighth. 01:02:00.410 --> 01:02:03.170 Multiply by 1/2 again, that's a 01:02:03.170 --> 01:02:08.846 16. Multiplied by 1/2 again, that's a 32nd. 01:02:09.640 --> 01:02:14.810 By half again that's a 64th by 1/2 again, that's 128. 01:02:16.080 --> 01:02:21.816 It's getting smaller, and if we do it enough times then it's 01:02:21.816 --> 01:02:24.684 going to head off till 0. 01:02:25.690 --> 01:02:30.530 What about a negative one? You might say, let's think about 01:02:30.530 --> 01:02:36.176 minus 1/2. Now multiplied by minus 1/2, it's a quarter. 01:02:36.176 --> 01:02:41.496 Multiply the quarter by minus 1/2. It's minus an eighth. 01:02:41.496 --> 01:02:46.816 Multiply again by minus 1/2. Well, that's plus a 16th. 01:02:46.816 --> 01:02:52.668 Multiply again by minus 1/2. That's minus a 32nd, so we're 01:02:52.668 --> 01:02:59.052 approaching 0, but where dotting about either side of 0 plus them 01:02:59.052 --> 01:03:03.840 were minus, then were plus then where mine is. 01:03:03.870 --> 01:03:08.562 We're getting nearer to zero each time, so again are to the 01:03:08.562 --> 01:03:13.645 power. N is going off to zero. What does that mean? It means 01:03:13.645 --> 01:03:19.119 that this some. Here we can have what we call a sum to Infinity. 01:03:19.119 --> 01:03:23.811 Sometimes it's just written with an S and sometimes it's got a 01:03:23.811 --> 01:03:25.766 little Infinity sign on it. 01:03:26.480 --> 01:03:32.822 What that tells us? Because this art of the end is going off to 01:03:32.822 --> 01:03:40.070 0 then it's a times by one over 1 minus R and that's our sum to 01:03:40.070 --> 01:03:42.788 Infinity. In other words, we can 01:03:42.788 --> 01:03:48.934 add up. An infinite number of terms for a geometric 01:03:48.934 --> 01:03:55.772 progression provided. The common ratio is between one and minus 01:03:55.772 --> 01:04:02.801 one, so let's have a look at an example. Supposing we've 01:04:02.801 --> 01:04:04.718 got this row. 01:04:04.750 --> 01:04:12.120 Metric progression. Well, first term is one 01:04:12.120 --> 01:04:16.524 now a common ratio is 1/3. 01:04:17.710 --> 01:04:19.796 And what does this come to when 01:04:19.796 --> 01:04:26.460 we add up? As many terms as we can, what's the sum to 01:04:26.460 --> 01:04:32.652 Infinity? We know the formula that's a over 1 minus R, so 01:04:32.652 --> 01:04:39.360 let's put the numbers in this one for a over 1 - 1/3. 01:04:40.460 --> 01:04:46.323 So the one on tops OK and the one minus third. Well that's 01:04:46.323 --> 01:04:52.186 2/3, and if we're dividing by a fraction then we invert it and 01:04:52.186 --> 01:04:57.147 multiply. So altogether that would come to three over 2, so 01:04:57.147 --> 01:04:59.853 it's very easy formula to use. 01:05:01.510 --> 01:05:04.246 Finally, just let's recap for a 01:05:04.246 --> 01:05:06.710 geometric progression. A. 01:05:07.360 --> 01:05:10.760 Is the first term. 01:05:10.760 --> 01:05:13.480 Aw. 01:05:14.740 --> 01:05:16.309 Is the common. 01:05:17.140 --> 01:05:24.026 Ratio. So a geometric progression looks like 01:05:24.026 --> 01:05:31.436 AARA, R-squared, AR, cubed and the N Terminus series AR 01:05:31.436 --> 01:05:35.141 to the N minus one. 01:05:35.780 --> 01:05:41.060 And if we want to add up this sequence of numbers SN. 01:05:41.820 --> 01:05:49.628 Then that's a Times 1 minus R to the power N or over 1 minus R. 01:05:50.490 --> 01:05:55.352 And if we're lucky enough to have our between plus and 01:05:55.352 --> 01:05:59.772 minus one, sometimes that's written as the modulus of art 01:05:59.772 --> 01:06:04.634 is less than one. If we're lucky to have this condition, 01:06:04.634 --> 01:06:10.380 then we can get a sum to Infinity, which is a over 1 01:06:10.380 --> 01:06:11.264 minus R.