WEBVTT 00:00:01.630 --> 00:00:08.566 This video is all about Cigna notation. This is a concise and 00:00:08.566 --> 00:00:11.456 precise way of explaining long 00:00:11.456 --> 00:00:17.468 sums. We study some examples and some special cases and then will 00:00:17.468 --> 00:00:19.100 drive some general results. 00:00:20.100 --> 00:00:22.816 But first, let us look at these. 00:00:23.570 --> 00:00:26.300 In math we have to, sometimes 00:00:26.300 --> 00:00:30.970 some. A number of terms in the sequence, like these two. 00:00:31.820 --> 00:00:37.243 The first one, 1 + 2 + 3 + 4 + 5. It's the sum of the 00:00:37.243 --> 00:00:38.519 first five whole numbers. 00:00:39.770 --> 00:00:47.618 The second one, 1 + 4 + 9 + 16 + 25 + 36. It's the sum of 00:00:47.618 --> 00:00:49.798 this first 6 square numbers. 00:00:51.740 --> 00:00:56.670 Now, if we had a general sequence of numbers such 00:00:56.670 --> 00:00:57.656 as this. 00:00:59.040 --> 00:01:06.420 You want you to you three, and so on. We could write the sum as 00:01:06.420 --> 00:01:13.308 Air Sequel you one, plus you 2 plus you three an so off. Now 00:01:13.308 --> 00:01:21.180 this some goes on and on and on. But if we want to some and terms 00:01:21.180 --> 00:01:27.576 then we can write that the sum for end terms equals you one 00:01:27.576 --> 00:01:29.052 plus you too. 00:01:29.070 --> 00:01:32.720 Bless you 3 up to 00:01:32.720 --> 00:01:36.470 UN. And those are in terms. 00:01:37.250 --> 00:01:41.540 Now we can use the Sigma notation to write this more 00:01:41.540 --> 00:01:46.744 concisely. And Sigma comes from the Greek Capital Letter, which 00:01:46.744 --> 00:01:52.096 corresponds to the S in some. And it looks like this Sigma. 00:01:53.430 --> 00:01:58.842 And if we want to express this some using Sigma notation, we 00:01:58.842 --> 00:02:01.548 take the general term you are. 00:02:02.530 --> 00:02:08.928 And we say that we are summing all terms like you are from R 00:02:08.928 --> 00:02:10.756 equal 1 two N. 00:02:11.610 --> 00:02:16.540 And that equals are some SN which is written above. 00:02:17.200 --> 00:02:22.997 More generally, we might take values of our starting at any 00:02:22.997 --> 00:02:29.848 point rather than just at one and finishing it in so we can 00:02:29.848 --> 00:02:37.226 write Sigma from our equal A to be have you are and that means 00:02:37.226 --> 00:02:40.388 the sum of all the terms. 00:02:40.930 --> 00:02:48.080 Like you are starting with our equal, a two are equal, BA is 00:02:48.080 --> 00:02:49.730 the lower limit. 00:02:51.530 --> 00:02:55.400 And B is the upper limit. 00:02:56.330 --> 00:03:01.530 Now what I'd like to do is explore some examples 00:03:01.530 --> 00:03:03.610 using the Sigma Notation. 00:03:04.620 --> 00:03:10.698 So take for example this one. 00:03:11.340 --> 00:03:13.680 It's Sigma of R cubed. 00:03:14.330 --> 00:03:19.465 With our starting are equal 1 and ending with R equals 4. So 00:03:19.465 --> 00:03:24.600 the lower limit is are equal 1 and the upper limit is are 00:03:24.600 --> 00:03:30.130 equal. 4 will all we have to do is just expanded out and write 00:03:30.130 --> 00:03:34.870 it as along some using the values are equal 123 and ending 00:03:34.870 --> 00:03:36.450 at our equal 4. 00:03:37.220 --> 00:03:44.288 When RS1R cubed is 1 cubed, are equal 2 two cubed are 00:03:44.288 --> 00:03:50.178 equal 3 three cubed and are equal 4 four cubed. 00:03:51.250 --> 00:03:52.898 And we can just. 00:03:53.520 --> 00:03:59.052 Work the Zeit and add them together. 1 cubed, 12 cubed is 00:03:59.052 --> 00:04:05.967 it 3 cubed? Is 3 * 3 * 3 so that's 936-2074 cubed is 4 00:04:05.967 --> 00:04:13.343 * 4 which is 16 times by 4 which is 64 and when we Add all 00:04:13.343 --> 00:04:17.492 of those together you get a lovely answer 100. 00:04:19.080 --> 00:04:23.040 What about this one? It's similar, but instead of using R 00:04:23.040 --> 00:04:27.720 is the variable. You can use N. In fact, you can use any 00:04:27.720 --> 00:04:29.160 variable that you want. 00:04:29.810 --> 00:04:34.874 And in this case were summing terms like N squared for men 00:04:34.874 --> 00:04:39.516 starting from 2 to 5, going up in increments of one. 00:04:40.050 --> 00:04:43.038 So starting with an equal 2. 00:04:43.800 --> 00:04:45.960 And squared is 2 squared. 00:04:46.580 --> 00:04:48.040 Plus 00:04:49.370 --> 00:04:55.530 An equal 3 three squared. An equal 4 four squared and 00:04:55.530 --> 00:04:58.330 finishing with an equal 55 00:04:58.330 --> 00:05:04.113 squared. And all we have to do now is square these out and add 00:05:04.113 --> 00:05:08.524 them together 2 squares for three squared, 9 four squared 16 00:05:08.524 --> 00:05:10.529 and 5 squared is 25. 00:05:11.120 --> 00:05:14.574 And we can add these very simply. Foreign 16 is 20. 00:05:15.110 --> 00:05:21.545 9 and 25 is 3420 and 34 is 54. 00:05:22.490 --> 00:05:23.998 What could be simpler? 00:05:24.630 --> 00:05:30.520 Look at these two written using the Sigma notation, just 00:05:30.520 --> 00:05:35.821 slightly different from the previous ones. Remember we look 00:05:35.821 --> 00:05:38.177 at all the terms. 00:05:38.800 --> 00:05:43.672 Such as two to the K and we substitute in the values for K 00:05:43.672 --> 00:05:47.500 in increments of one starting with zero going up to five. 00:05:48.530 --> 00:05:55.982 So the first one 2 to the K when K is 0, is 2 to the North. Then 00:05:55.982 --> 00:06:03.434 we add 2 to the one +2 to the 2 + 2 to 3 + 2 to the 00:06:03.434 --> 00:06:05.918 4 + 2 to the five. 00:06:06.530 --> 00:06:10.202 Notice we've got six terms and the reason why we've got six 00:06:10.202 --> 00:06:13.568 terms is we started K at zero and not at one. 00:06:14.310 --> 00:06:21.707 And then we evaluate those two to the zero 1 + 2 plus 00:06:21.707 --> 00:06:29.104 force plus three 2 cubed, which is it +2 to 416. Let's do 00:06:29.104 --> 00:06:36.670 the 5:32. And when we add those up, 1 + 2 00:06:36.670 --> 00:06:41.220 is 347-1531. And then 63. 00:06:43.100 --> 00:06:45.850 Now look at this one. 00:06:46.100 --> 00:06:50.570 A little bit more involved, but quite straightforward if you 00:06:50.570 --> 00:06:55.934 follow the same steps that we've done before, we take the general 00:06:55.934 --> 00:07:00.851 term here and we substitute in values looking at where it 00:07:00.851 --> 00:07:06.662 starts from and where it ends. So R equals one is the lower 00:07:06.662 --> 00:07:12.473 limit, so we look at our general term and substituted are equal 1 00:07:12.473 --> 00:07:16.943 first and that will give us one times by two. 00:07:16.950 --> 00:07:20.846 Times by 1/2 added 00:07:20.846 --> 00:07:28.576 two. Now we substitute are equal to for the next one, so that's 00:07:28.576 --> 00:07:35.332 two are plus one is 3 + 1/2. An hour equals 3. 00:07:35.750 --> 00:07:38.040 3 plus One is 4. 00:07:38.590 --> 00:07:41.380 Plus 1/2 an. 00:07:41.920 --> 00:07:48.839 And I forgot my arm, but my are in this case is 4 and we add 1 00:07:48.839 --> 00:07:55.351 to it which is 5 and continue on in the same then until we get to 00:07:55.351 --> 00:08:01.049 our upper limit our equals 6. So that's a half six times by 7. 00:08:01.760 --> 00:08:05.881 And we can evaluate this side, but we can do a lot of 00:08:05.881 --> 00:08:09.685 canceling on the way before we do. And if the adding two 00:08:09.685 --> 00:08:13.489 cancers with the two here and here, 2 cancers with the four 00:08:13.489 --> 00:08:17.927 to give us two here 2 cancers with this forward to give us 2 00:08:17.927 --> 00:08:21.731 two counters with this six to give us three and two counters 00:08:21.731 --> 00:08:23.950 with this six to give us 3. 00:08:25.410 --> 00:08:31.052 And then we evaluate eyes adding together terms. So this one is 1 00:08:31.052 --> 00:08:38.278 plus. And we're left with 1 * 1 * 3, which is 3 00:08:38.278 --> 00:08:43.318 + 1 * 3 * 2, which is 6 plus. 00:08:44.220 --> 00:08:46.386 2 * 5 which is 10. 00:08:47.210 --> 00:08:52.677 Plus 3 * 5 which is 15 and the last 121. 00:08:53.250 --> 00:08:58.020 Now if you look at these numbers, these are quite 00:08:58.020 --> 00:09:02.313 special numbers. There's a name given to them, they 00:09:02.313 --> 00:09:03.744 either triangular numbers. 00:09:04.840 --> 00:09:08.788 And you should be able to recognize these the same way as 00:09:08.788 --> 00:09:11.749 you would recognize square numbers or prime numbers, or 00:09:11.749 --> 00:09:17.221 order even numbers. I'm finally what would like to do is just 00:09:17.221 --> 00:09:22.947 add these altogether. 1 + 3 + 6 is 10 + 10 is 20. 00:09:24.520 --> 00:09:31.060 20 + 1535 thirty six and then add the 20 is 56. 00:09:32.250 --> 00:09:34.506 So we get to some 56. 00:09:35.350 --> 00:09:38.020 Now, what 00:09:38.020 --> 00:09:44.436 about this? We've got our general term 2 to 00:09:44.436 --> 00:09:50.804 the K and we want to start with K equal 1 decay equal an if we 00:09:50.804 --> 00:09:55.580 write it out substituting in our values for K starting with K 00:09:55.580 --> 00:10:03.116 equal 1. Our first time will be 2 to the one plus then case two 00:10:03.116 --> 00:10:10.348 would be 2 to the 2 + 2 to 3 + 2 to the 4th and 00:10:10.348 --> 00:10:14.274 so on. Until we get to two to the N. 00:10:15.880 --> 00:10:19.354 Now we can't evaluate this to a finite number, 00:10:19.354 --> 00:10:23.214 because at this stage we don't know what Aaron is. 00:10:25.960 --> 00:10:33.144 And what if we have these two expressions? 00:10:33.990 --> 00:10:38.401 But still using the Sigma notation, but in this case are 00:10:38.401 --> 00:10:42.010 general term involves a negative sign in both cases. 00:10:42.610 --> 00:10:46.029 So we'll just take a little bit of time and carefully work the 00:10:46.029 --> 00:10:50.832 might. Because it's dead easy to go wrong in these sorts of ones, 00:10:50.832 --> 00:10:55.326 so take a little bit of time and you will hopefully work them out 00:10:55.326 --> 00:10:57.252 OK using the same steps that 00:10:57.252 --> 00:11:01.390 we've done before. In this particular example, our general 00:11:01.390 --> 00:11:07.620 term is negative, one raised to the part of our and R equals 1 00:11:07.620 --> 00:11:13.850 to 4. So the first term will be negative. One reason the part of 00:11:13.850 --> 00:11:19.773 one. Plus negative one race to the power of 2 plus negative 00:11:19.773 --> 00:11:24.646 one. Raise the pirate three and finally negative one raised to 00:11:24.646 --> 00:11:26.418 the power of 4. 00:11:27.260 --> 00:11:32.525 And then all we have to do is multiply these item, making sure 00:11:32.525 --> 00:11:37.385 that we get our signs correct, negative one raise the part of 00:11:37.385 --> 00:11:42.245 one is negative, one negative one raised to the part of two 00:11:42.245 --> 00:11:44.270 that's squared is positive one. 00:11:44.950 --> 00:11:47.038 Negative 1 cubed. 00:11:47.640 --> 00:11:52.008 That will give negative one and negative one is part of four. 00:11:52.008 --> 00:11:53.464 That'll give positive one. 00:11:54.560 --> 00:11:56.549 So in fact. 00:11:57.470 --> 00:12:02.891 Are some work site to be O negative 1 + 1 zero plus 00:12:02.891 --> 00:12:04.976 negative 1 + 1 zero? 00:12:05.700 --> 00:12:07.206 Quite a surprise, in fact, in 00:12:07.206 --> 00:12:10.790 fact. Now look at this one. 00:12:11.440 --> 00:12:16.632 Again, take care. Our general term is negative one over K 00:12:16.632 --> 00:12:22.768 squared and we have to use values of K from one to three. 00:12:23.650 --> 00:12:31.534 So we start with K equal 1. That's negative one over 1 00:12:31.534 --> 00:12:39.418 squared plus negative one over 2 squared plus negative one over 3 00:12:39.418 --> 00:12:45.659 squared. And then again, taking care with the negative signs. 00:12:45.659 --> 00:12:51.610 Square these values out negative one over one is negative, 1 00:12:51.610 --> 00:12:54.856 squared will give us one plus. 00:12:55.470 --> 00:13:00.606 Negative one over 2 all squared that would be positive and is 00:13:00.606 --> 00:13:03.174 1/2 * 1/2, which is quarter. 00:13:03.940 --> 00:13:07.603 Then added two negative one over 3 squared. Again, that's going 00:13:07.603 --> 00:13:12.598 to be positive, and it's going to be one over 3 * 1 over 3, 00:13:12.598 --> 00:13:14.929 which is one of her and I. 00:13:15.690 --> 00:13:20.019 We can add these together, adding the two fractions 00:13:20.019 --> 00:13:20.500 together. 00:13:21.560 --> 00:13:23.768 The common denominators 36. 00:13:24.310 --> 00:13:31.465 And it will be 9 + 4 and you've got your one there. So my 00:13:31.465 --> 00:13:37.666 final answer is one and 13 over 36. Quite a grotty answer, but. 00:13:38.180 --> 00:13:42.860 We've been able to workout what seemed to be quite a complicated 00:13:42.860 --> 00:13:44.030 Sigma notation some. 00:13:44.660 --> 00:13:46.646 And we finally got an answer. 00:13:47.660 --> 00:13:53.344 But what if we had along some and we wanted to write it in 00:13:53.344 --> 00:13:55.856 Sigma Notation? How would we do 00:13:55.856 --> 00:14:01.768 it? Well, why don't we first look at the two sums that we 00:14:01.768 --> 00:14:03.010 started off with? 00:14:03.020 --> 00:14:10.050 The first one was 1 + 2 + 3 + 00:14:10.050 --> 00:14:12.159 4 + 5. 00:14:12.930 --> 00:14:17.130 This one is quite an easy one to start off with because it's the 00:14:17.130 --> 00:14:21.147 sum. What type of terms? Well, it's a dead easy term because 00:14:21.147 --> 00:14:25.528 they are adding one on each time. So if we take the general 00:14:25.528 --> 00:14:26.539 term to BK. 00:14:28.040 --> 00:14:34.449 K starts the lower limit one and ends with the upper limit 5. 00:14:35.750 --> 00:14:39.240 So there's no problem with this one. This is very 00:14:39.240 --> 00:14:41.884 straightforward. But look at the 00:14:41.884 --> 00:14:46.720 second one. Again, it's not too bad because we notice and we 00:14:46.720 --> 00:14:48.650 said before that these are 00:14:48.650 --> 00:14:53.972 square numbers. So we could actually write these 00:14:53.972 --> 00:15:00.642 as one squared +2 squared, +3 squared, +4 squared, 5 00:15:00.642 --> 00:15:03.310 squared, and six squared. 00:15:04.740 --> 00:15:08.046 Once we've done this, are Sigma 00:15:08.046 --> 00:15:15.377 notation some? Is dead easy because we could easily see our 00:15:15.377 --> 00:15:18.935 general term must be K squared. 00:15:20.450 --> 00:15:23.411 And K must start with cake, will 00:15:23.411 --> 00:15:26.628 one. Takei equals 6. 00:15:27.240 --> 00:15:32.784 But what about these 00:15:32.784 --> 00:15:39.920 ones? This is a long some. 00:15:41.370 --> 00:15:46.290 What's different about this? It's got fractions and it also 00:15:46.290 --> 00:15:47.766 has alternating signs. 00:15:48.910 --> 00:15:52.995 Negative. Positive negative positive negative positive 00:15:52.995 --> 00:15:56.005 silver. Now the trick in this 00:15:56.005 --> 00:16:01.630 case. Is to rewrite all of these in terms of fractions. 00:16:02.740 --> 00:16:06.044 And then use negative 1 to a 00:16:06.044 --> 00:16:12.194 power. So if we look at this, we can rewrite this as negative one 00:16:12.194 --> 00:16:15.150 over 1. Plus 1/2. 00:16:16.830 --> 00:16:20.590 Minus one over 3 + 00:16:20.590 --> 00:16:25.459 1/4 plus. And So what to one over 100? 00:16:27.470 --> 00:16:33.302 Now we have to deal with the signs alternating. 00:16:34.620 --> 00:16:38.734 And if we think back we did have signs alternative before. 00:16:39.290 --> 00:16:42.782 And that's to do with negative numbers. Now we don't want to 00:16:42.782 --> 00:16:45.983 change the value of the fraction, we just want to change 00:16:45.983 --> 00:16:51.934 the sign. So if I rewrite this as negative one. 00:16:52.570 --> 00:16:54.720 Times by one over 1. 00:16:55.760 --> 00:16:59.710 Plus Negative one. 00:16:59.710 --> 00:17:01.398 The power of 2. 00:17:02.750 --> 00:17:04.019 One over 2. 00:17:05.010 --> 00:17:10.246 Plus negative 1 to the power of 3 one over 3. 00:17:10.780 --> 00:17:15.620 Plus negative 1 to the power of 4 one over 4. 00:17:17.230 --> 00:17:23.500 Time 2 plus negative one to the 101 over 100. 00:17:24.030 --> 00:17:25.968 Have we got the same sum? 00:17:26.890 --> 00:17:29.669 One yes, because look at each term. 00:17:31.760 --> 00:17:34.807 Negative 1 * 1 over one is negative, one over 1. 00:17:36.070 --> 00:17:40.965 Negative 1 squared will be positive 1 * 1/2, which will 00:17:40.965 --> 00:17:43.190 give us our plus 1/2. 00:17:43.850 --> 00:17:48.734 Negative 1 cubed will be negative one times by 1/3 which 00:17:48.734 --> 00:17:54.062 is negative 1/3, so that's where we get the negative 1/3 negative 00:17:54.062 --> 00:17:56.726 1 to the power of 4. 00:17:57.340 --> 00:18:01.825 Is positive one times by 1/4 gives us our plus one over 4. 00:18:02.940 --> 00:18:06.570 So rewriting our sequence that we started off with here. 00:18:07.360 --> 00:18:08.809 Into this format. 00:18:09.620 --> 00:18:13.820 That then helps us to write it in Sigma Notation. 00:18:15.430 --> 00:18:19.705 Because we can easily see that this is Sigma. 00:18:21.120 --> 00:18:24.950 We know it's negative one. 00:18:25.180 --> 00:18:31.879 To the power and the power is always the number underneath 00:18:31.879 --> 00:18:37.570 the fraction. So we'll say that our fraction is one over K 00:18:37.570 --> 00:18:42.764 because we start with K equal 1 and we raise negative 1 to the 00:18:42.764 --> 00:18:43.877 power of K. 00:18:44.570 --> 00:18:49.180 And to complete our Sigma notation for this long some. 00:18:50.050 --> 00:18:53.600 We start with the lower limit of K equal 1. 00:18:54.130 --> 00:18:57.105 And the upper limit K equal 100. 00:18:57.930 --> 00:19:00.926 So you can easily see that this 00:19:00.926 --> 00:19:03.429 notation. Is very concise. 00:19:04.100 --> 00:19:08.434 And I really lovely way of writing that big long some. 00:19:09.600 --> 00:19:16.950 Now I'd like to move on some special cases using 00:19:16.950 --> 00:19:21.744 Sigma notation. The first special case is when you have to 00:19:21.744 --> 00:19:24.290 sum a constant. Like this? 00:19:24.840 --> 00:19:29.351 If we have Sigma of three from K equal 1 to 5, what 00:19:29.351 --> 00:19:30.739 does that actually mean? 00:19:31.950 --> 00:19:33.940 Well, we work at night. 00:19:34.460 --> 00:19:41.740 We take each term for cake with one up to K equal 5. 00:19:42.660 --> 00:19:49.764 Each time remains the same. The matter whether cake was one K 00:19:49.764 --> 00:19:56.868 equals 2 cakes, 3, four, or five. So we actually just get 00:19:56.868 --> 00:19:59.828 the sum of 5 threes. 00:20:02.430 --> 00:20:04.380 Which is 15th. 00:20:06.080 --> 00:20:12.262 So with five terms, five times the constant to make 15. 00:20:13.170 --> 00:20:18.994 So we can get a general result from that if instead of three we 00:20:18.994 --> 00:20:20.658 have a constant C. 00:20:22.290 --> 00:20:26.520 And we take our K from one to N. 00:20:27.690 --> 00:20:32.790 What does this work? I to be? Well, we do it in the same way. 00:20:32.800 --> 00:20:35.062 We workout are some using cake 00:20:35.062 --> 00:20:39.580 with one. 2, three, and so on up to K equal an. 00:20:40.280 --> 00:20:41.600 And evaluate it. 00:20:42.100 --> 00:20:48.832 So can equal 1 is say K Equal 2C and so on. 00:20:50.780 --> 00:20:57.854 Up to. The last see such that we have and terms. 00:20:58.780 --> 00:21:05.176 Because we've got N lots of see this, some can be simplified 00:21:05.176 --> 00:21:08.374 dead easily to an times C. 00:21:10.620 --> 00:21:15.360 So this is a general result which is very useful when using 00:21:15.360 --> 00:21:18.915 Sigma notation. If you have to sum a constant. 00:21:19.700 --> 00:21:26.434 From cable one to N, then the value is and it is a finite 00:21:26.434 --> 00:21:28.839 value and times by C. 00:21:29.420 --> 00:21:30.988 What could be easier? 00:21:31.820 --> 00:21:34.668 But what about this? 00:21:35.910 --> 00:21:42.950 What if you had to sum a variable like this 3K? 00:21:43.480 --> 00:21:45.688 Can this be written more easily? 00:21:46.300 --> 00:21:48.390 Well, we'll take this particular 00:21:48.390 --> 00:21:53.924 example. 3K some 4K. Will want to four. 00:21:54.540 --> 00:22:01.400 So will sub in cake. Will 1 first, then K equal 2 then K 00:22:01.400 --> 00:22:07.280 equals 3 and then K equals 4 and we get that some. 00:22:07.820 --> 00:22:12.318 We notice that three is a factor if each one of these terms. 00:22:12.850 --> 00:22:20.172 So we factorize IR 3 and we get three bracket 1 + 2 + 00:22:20.172 --> 00:22:21.741 3 + 4. 00:22:23.220 --> 00:22:29.688 Now that is equal to three times by. If you want to add these 00:22:29.688 --> 00:22:34.308 altogether, that's three 610. So the answer is actually 30. 00:22:35.070 --> 00:22:39.102 Now I'm not so much interested in the answer, but in the 00:22:39.102 --> 00:22:43.990 process. At anyone stage, can you spot something that we know 00:22:43.990 --> 00:22:49.110 already? Well, look at this second line. You've got 3 times 00:22:49.110 --> 00:22:52.454 by 1 + 2 + 3 + 4. 00:22:53.770 --> 00:23:00.166 That's the addition of K&K is one 2, three and four, and we 00:23:00.166 --> 00:23:01.642 can rewrite this. 00:23:02.410 --> 00:23:06.174 Using Sigma notation that 00:23:06.174 --> 00:23:08.056 is 3. 00:23:08.580 --> 00:23:15.460 Sigma. Values K from K equals 00:23:15.460 --> 00:23:16.270 124. 00:23:17.500 --> 00:23:23.429 So our expression that we started off with Sigma of three 00:23:23.429 --> 00:23:29.897 K equals 1 to 4K was one to four equals 3 times. 00:23:30.470 --> 00:23:36.482 Sigma of K from cable one to four. So effectively we've taken 00:23:36.482 --> 00:23:41.492 out with factorized at three from our first expression, and 00:23:41.492 --> 00:23:44.498 we brought it outside the Sigma 00:23:44.498 --> 00:23:48.200 notation. So more generally. 00:23:48.820 --> 00:23:56.506 We can take that if we've got the Sigma of CK from K equals 00:23:56.506 --> 00:23:58.153 1 to N. 00:23:58.940 --> 00:24:00.160 What would that equal? 00:24:00.680 --> 00:24:07.610 Wow, quite easy. All we have to do is that we see that that is. 00:24:08.280 --> 00:24:14.869 See times one plus C times 2 plus C times 3. 00:24:16.020 --> 00:24:17.109 Right up to. 00:24:17.680 --> 00:24:20.571 See times an. We've got M terms 00:24:20.571 --> 00:24:23.665 here. We can factorize by the 00:24:23.665 --> 00:24:30.980 sea. And we get 1 + 2 + 3 plus up 00:24:30.980 --> 00:24:37.190 to N and then we can rewrite this using Sigma 00:24:37.190 --> 00:24:44.021 notation, that is C Sigma of care from K equals 1 00:24:44.021 --> 00:24:45.263 to N. 00:24:46.440 --> 00:24:47.548 And as I say. 00:24:48.420 --> 00:24:53.337 Quite straightforward, when you've got a Sigma of C times a 00:24:53.337 --> 00:24:57.807 variable. See being a constant, you can take the constant 00:24:57.807 --> 00:25:03.171 outside the Sigma notation and you left with C times by the 00:25:03.171 --> 00:25:06.747 Sigma of K from K equal 1 to 00:25:06.747 --> 00:25:11.733 N. That's another very useful result that you should remember 00:25:11.733 --> 00:25:13.401 when using Sigma notation. 00:25:14.190 --> 00:25:20.185 But what if we had this expression? The sum of the 00:25:20.185 --> 00:25:26.725 variable K plus two where K takes the lower limit one to 00:25:26.725 --> 00:25:33.810 four? What does that work? I'd be? Can we write it more concert 00:25:33.810 --> 00:25:35.990 concisely and easily well? 00:25:36.690 --> 00:25:40.988 We do it stage by stage, step by step, putting in values for K 00:25:40.988 --> 00:25:42.830 from K equal 1 to 4. 00:25:43.480 --> 00:25:47.310 So the first time case one says 1 + 2. 00:25:48.720 --> 00:25:53.780 Plus second, her K equals 2, so that's 2 + 2. 00:25:54.470 --> 00:26:01.370 The third term is 3 + 2 and the fourth term is 4 + 2. 00:26:02.580 --> 00:26:07.926 Now what I want to do next is rearrange these values. 00:26:09.040 --> 00:26:15.823 If I rearrange such that I have the addition of 1 + 2 + 3 + 4 00:26:15.823 --> 00:26:23.114 first. And then see what we've got left. We've got 00:26:23.114 --> 00:26:29.974 add 2 four times, so that is 4 * 2. 00:26:31.420 --> 00:26:37.630 And then we can easily evaluate this expression 1 + 2 + 3 + 4 00:26:37.630 --> 00:26:41.356 is 10 + 4 two 38. So the answer 00:26:41.356 --> 00:26:46.898 is 18. But as I said before, we're not so much concerned with 00:26:46.898 --> 00:26:51.910 the answer, but in the process, can we look and see at any line? 00:26:52.590 --> 00:26:54.948 Can we use the Sigma notation 00:26:54.948 --> 00:26:58.050 more easily? Well, if we look at 00:26:58.050 --> 00:27:03.686 this line. We've seen part of this before. 00:27:04.370 --> 00:27:07.506 1 + 2 + 3 + 4. 00:27:08.640 --> 00:27:14.305 Can be written more easily using Sigma notation as Sigma. Care 00:27:14.305 --> 00:27:17.395 from K equals 1 to 4. 00:27:18.080 --> 00:27:23.337 Plus then we've got 4 * 2. 00:27:25.430 --> 00:27:31.022 So looking back at what we started with, we've got Sigma of 00:27:31.022 --> 00:27:38.012 K plus two from K equals one to four and it expanded I to this 00:27:38.012 --> 00:27:44.536 Sigma notation Sigma of can at all, from cable one to 4 + 4 00:27:44.536 --> 00:27:50.594 * 2 or two is the constant that was added to the care. 00:27:52.410 --> 00:27:54.828 So we can generalize this result. 00:27:56.950 --> 00:28:01.290 If we had Sigma of. 00:28:02.340 --> 00:28:08.820 G of K function of K plus a constant C. 00:28:09.320 --> 00:28:16.509 As our variable and we want to take care from one to N. 00:28:17.910 --> 00:28:23.430 We can use the pattern that we spotted here to write this 00:28:23.430 --> 00:28:27.110 out using Sigma notation plus some other value. 00:28:28.410 --> 00:28:32.070 The first part is Sigma. 00:28:32.760 --> 00:28:38.545 Off the function that we started off with in our variable and the 00:28:38.545 --> 00:28:40.325 function is GF care. 00:28:41.010 --> 00:28:48.270 And we see that it should be from K to the upper limit from K 00:28:48.270 --> 00:28:54.548 equal 1. Takei equal the upper limit in this case is for, but 00:28:54.548 --> 00:28:57.334 in our general case it's K equal 00:28:57.334 --> 00:29:00.720 N. Says K equal 1 to an. 00:29:01.320 --> 00:29:08.384 Plus Now the next part, remember it was the upper limit by the 00:29:08.384 --> 00:29:13.688 number of terms times by the constant that you've had in your 00:29:13.688 --> 00:29:15.898 variable. Well, look here. Are 00:29:15.898 --> 00:29:22.748 constantly see. And our upper variable is an. 00:29:23.280 --> 00:29:25.870 So it's End Times C. 00:29:26.780 --> 00:29:30.255 So this Sigma notation can 00:29:30.255 --> 00:29:37.540 be expanded. Two Sigma GF K from K Equal 1M Plus M 00:29:37.540 --> 00:29:38.590 Times C. 00:29:40.210 --> 00:29:44.746 And we can even generalize further using previous results 00:29:44.746 --> 00:29:49.786 if we took Sigma from K equals 1 to N. 00:29:50.370 --> 00:29:54.210 Of a GF K 00:29:54.210 --> 00:29:57.820 Plus C. As our 00:29:57.820 --> 00:30:01.368 variable. We can expand that 00:30:01.368 --> 00:30:07.750 out. Looking at this time 1st and then dealing with the 00:30:07.750 --> 00:30:12.997 constant now, this term is a multiple a constant times by 00:30:12.997 --> 00:30:18.721 your GF K so we can bring that outside the Sigma notation. 00:30:18.810 --> 00:30:26.766 And say it's a Times by Sigma GfK from K equals 1 00:30:26.766 --> 00:30:28.092 to N. 00:30:29.560 --> 00:30:30.630 Plus 00:30:32.670 --> 00:30:38.065 Just what's left to do with the sea and we know when that's 00:30:38.065 --> 00:30:41.385 inside the bracket when we expand it out. 00:30:42.070 --> 00:30:44.416 We're left with end times C. 00:30:45.730 --> 00:30:46.810 What could be easier? 00:30:47.560 --> 00:30:51.260 All right? But 00:30:51.260 --> 00:30:57.566 what if? We wanted to use Sigma notation and the 00:30:57.566 --> 00:31:03.350 variable we wanted to add together is K Plus K squared IE 00:31:03.350 --> 00:31:05.278 two functions of K. 00:31:07.060 --> 00:31:11.053 What would we get when we write out this long some? 00:31:11.660 --> 00:31:14.480 Start simply by substituting in 00:31:14.480 --> 00:31:20.683 for K. Case one to begin with. So the first term is 1 + 1 00:31:20.683 --> 00:31:28.043 squared. Second time is 2 + 2 squared. Third time is 3 + 3 00:31:28.043 --> 00:31:33.224 squared, and that's our final term because the upper limit for 00:31:33.224 --> 00:31:34.637 K is 3. 00:31:35.300 --> 00:31:41.528 Now we'll rearrange the way we did before to make our work 00:31:41.528 --> 00:31:46.718 easier. Will take the 1 + 2 + 3 together. 00:31:49.040 --> 00:31:53.473 And then we'll take the one squared and the two squared 00:31:53.473 --> 00:31:56.697 and the three squared bits that are left. 00:31:57.860 --> 00:32:02.684 And you can see why I've done that, because here we're adding 00:32:02.684 --> 00:32:05.096 our constant case and here we're 00:32:05.096 --> 00:32:07.828 adding. Our case squareds. 00:32:08.750 --> 00:32:14.282 When we add these together, we just add them up and multiply 00:32:14.282 --> 00:32:20.736 out using the square sign you get 1 + 2 + 3 is 6, 00:32:20.736 --> 00:32:26.729 one squared is 1, two squared is 4 three squared 9. Add back 00:32:26.729 --> 00:32:31.339 together that gives you fourteen. 6 + 14 is 20. 00:32:33.160 --> 00:32:36.286 But look at this second line. 00:32:37.420 --> 00:32:42.685 We can simplify this second line using Sigma notation. 00:32:43.510 --> 00:32:48.762 And it's using Sigma notation that we've done before. 1 + 2 + 00:32:48.762 --> 00:32:54.418 3 is the same as Sigma of care from K equals 1 to 3. 00:32:55.350 --> 00:32:56.100 Plus 00:32:57.480 --> 00:33:02.560 1 squared, +2 squared, +3 squared, that's dead easy. It's 00:33:02.560 --> 00:33:09.164 the sum of all variables, K squared from K equals 1 to 3. 00:33:10.320 --> 00:33:14.159 Now look at what would started and look what we've got. 00:33:14.870 --> 00:33:21.580 We have broken this Sigma notation up into two sums to 00:33:21.580 --> 00:33:28.540 Sigma sums. So here we've got the Sigma offer variable 00:33:28.540 --> 00:33:30.490 plus another variable. 00:33:31.420 --> 00:33:34.000 And with split it up to. 00:33:34.720 --> 00:33:39.364 Sigma of one variable added to the Sigma of the other variable. 00:33:40.420 --> 00:33:42.330 We've used the distributive law. 00:33:43.350 --> 00:33:46.880 So to write this more 00:33:46.880 --> 00:33:52.133 generally. We can say that the sum of a function of care. 00:33:53.140 --> 00:33:57.466 Added to another function of K. 00:33:58.950 --> 00:34:04.610 From K equals 1 to N must equal the sum. 00:34:05.900 --> 00:34:13.284 Two separate sums. The sum of JFK from K equals 1 to M 00:34:13.284 --> 00:34:19.755 Plus Sigma. FFK from K equals 1 to add. 00:34:21.580 --> 00:34:26.546 And in fact, we can continue this on if we added on other 00:34:26.546 --> 00:34:28.074 functions within our variable 00:34:28.074 --> 00:34:34.606 here. Then all we have to do is add on another Sigma of 00:34:34.606 --> 00:34:35.965 that particular function. 00:34:37.980 --> 00:34:43.047 Finally, here's a real life example using Sigma notation. 00:34:46.440 --> 00:34:50.388 Take for example if we want to find the mean of a set of 00:34:50.388 --> 00:34:52.926 numbers. That could be the marks in a test. 00:34:54.230 --> 00:34:58.790 If we want to find the main what we want to do is we have to 00:34:58.790 --> 00:34:59.930 workout the total sum. 00:35:01.500 --> 00:35:04.530 Divided by the number of values. 00:35:05.580 --> 00:35:10.816 And that's what the main is. Well, that's actually what we've 00:35:10.816 --> 00:35:15.576 been doing in the previous examples. We have been finding 00:35:15.576 --> 00:35:21.764 the total sum, and we have been looking at the number of values. 00:35:21.764 --> 00:35:27.952 So if I take this example, just say we had the marks 23456. 00:35:29.210 --> 00:35:34.150 And we want to find the mean of those marks. The main will 00:35:34.150 --> 00:35:41.924 equal. 2 + 3 + 4 + 5 + 6 and we'd have to divide 00:35:41.924 --> 00:35:48.112 it by the number of values, which is 5 when we work that 00:35:48.112 --> 00:35:51.444 out. That is 5 + 510, another 00:35:51.444 --> 00:35:57.056 10. 20 over 5 which works out to be the value 4. 00:35:57.910 --> 00:35:59.650 But more generally. 00:36:00.210 --> 00:36:07.280 If you've got a set of marks, say XI, we can write the main 00:36:07.280 --> 00:36:13.845 in terms of Sigma Notation. It's the total sum of all the marks. 00:36:14.380 --> 00:36:15.829 Such as Zhao. 00:36:17.630 --> 00:36:20.090 From I equals 1 to N. 00:36:20.830 --> 00:36:25.940 And then we want to divide it by the number of values. Now we 00:36:25.940 --> 00:36:27.765 know there are N Marks. 00:36:28.570 --> 00:36:35.738 So we want to pie by one over N, so the main is equal to one 00:36:35.738 --> 00:36:41.114 over an Sigma X of I from I equal 1 to M.