1 00:00:01,630 --> 00:00:08,566 This video is all about Cigna notation. This is a concise and 2 00:00:08,566 --> 00:00:11,456 precise way of explaining long 3 00:00:11,456 --> 00:00:17,468 sums. We study some examples and some special cases and then will 4 00:00:17,468 --> 00:00:19,100 drive some general results. 5 00:00:20,100 --> 00:00:22,816 But first, let us look at these. 6 00:00:23,570 --> 00:00:26,300 In math we have to, sometimes 7 00:00:26,300 --> 00:00:30,970 some. A number of terms in the sequence, like these two. 8 00:00:31,820 --> 00:00:37,243 The first one, 1 + 2 + 3 + 4 + 5. It's the sum of the 9 00:00:37,243 --> 00:00:38,519 first five whole numbers. 10 00:00:39,770 --> 00:00:47,618 The second one, 1 + 4 + 9 + 16 + 25 + 36. It's the sum of 11 00:00:47,618 --> 00:00:49,798 this first 6 square numbers. 12 00:00:51,740 --> 00:00:56,670 Now, if we had a general sequence of numbers such 13 00:00:56,670 --> 00:00:57,656 as this. 14 00:00:59,040 --> 00:01:06,420 You want you to you three, and so on. We could write the sum as 15 00:01:06,420 --> 00:01:13,308 Air Sequel you one, plus you 2 plus you three an so off. Now 16 00:01:13,308 --> 00:01:21,180 this some goes on and on and on. But if we want to some and terms 17 00:01:21,180 --> 00:01:27,576 then we can write that the sum for end terms equals you one 18 00:01:27,576 --> 00:01:29,052 plus you too. 19 00:01:29,070 --> 00:01:32,720 Bless you 3 up to 20 00:01:32,720 --> 00:01:36,470 UN. And those are in terms. 21 00:01:37,250 --> 00:01:41,540 Now we can use the Sigma notation to write this more 22 00:01:41,540 --> 00:01:46,744 concisely. And Sigma comes from the Greek Capital Letter, which 23 00:01:46,744 --> 00:01:52,096 corresponds to the S in some. And it looks like this Sigma. 24 00:01:53,430 --> 00:01:58,842 And if we want to express this some using Sigma notation, we 25 00:01:58,842 --> 00:02:01,548 take the general term you are. 26 00:02:02,530 --> 00:02:08,928 And we say that we are summing all terms like you are from R 27 00:02:08,928 --> 00:02:10,756 equal 1 two N. 28 00:02:11,610 --> 00:02:16,540 And that equals are some SN which is written above. 29 00:02:17,200 --> 00:02:22,997 More generally, we might take values of our starting at any 30 00:02:22,997 --> 00:02:29,848 point rather than just at one and finishing it in so we can 31 00:02:29,848 --> 00:02:37,226 write Sigma from our equal A to be have you are and that means 32 00:02:37,226 --> 00:02:40,388 the sum of all the terms. 33 00:02:40,930 --> 00:02:48,080 Like you are starting with our equal, a two are equal, BA is 34 00:02:48,080 --> 00:02:49,730 the lower limit. 35 00:02:51,530 --> 00:02:55,400 And B is the upper limit. 36 00:02:56,330 --> 00:03:01,530 Now what I'd like to do is explore some examples 37 00:03:01,530 --> 00:03:03,610 using the Sigma Notation. 38 00:03:04,620 --> 00:03:10,698 So take for example this one. 39 00:03:11,340 --> 00:03:13,680 It's Sigma of R cubed. 40 00:03:14,330 --> 00:03:19,465 With our starting are equal 1 and ending with R equals 4. So 41 00:03:19,465 --> 00:03:24,600 the lower limit is are equal 1 and the upper limit is are 42 00:03:24,600 --> 00:03:30,130 equal. 4 will all we have to do is just expanded out and write 43 00:03:30,130 --> 00:03:34,870 it as along some using the values are equal 123 and ending 44 00:03:34,870 --> 00:03:36,450 at our equal 4. 45 00:03:37,220 --> 00:03:44,288 When RS1R cubed is 1 cubed, are equal 2 two cubed are 46 00:03:44,288 --> 00:03:50,178 equal 3 three cubed and are equal 4 four cubed. 47 00:03:51,250 --> 00:03:52,898 And we can just. 48 00:03:53,520 --> 00:03:59,052 Work the Zeit and add them together. 1 cubed, 12 cubed is 49 00:03:59,052 --> 00:04:05,967 it 3 cubed? Is 3 * 3 * 3 so that's 936-2074 cubed is 4 50 00:04:05,967 --> 00:04:13,343 * 4 which is 16 times by 4 which is 64 and when we Add all 51 00:04:13,343 --> 00:04:17,492 of those together you get a lovely answer 100. 52 00:04:19,080 --> 00:04:23,040 What about this one? It's similar, but instead of using R 53 00:04:23,040 --> 00:04:27,720 is the variable. You can use N. In fact, you can use any 54 00:04:27,720 --> 00:04:29,160 variable that you want. 55 00:04:29,810 --> 00:04:34,874 And in this case were summing terms like N squared for men 56 00:04:34,874 --> 00:04:39,516 starting from 2 to 5, going up in increments of one. 57 00:04:40,050 --> 00:04:43,038 So starting with an equal 2. 58 00:04:43,800 --> 00:04:45,960 And squared is 2 squared. 59 00:04:46,580 --> 00:04:48,040 Plus 60 00:04:49,370 --> 00:04:55,530 An equal 3 three squared. An equal 4 four squared and 61 00:04:55,530 --> 00:04:58,330 finishing with an equal 55 62 00:04:58,330 --> 00:05:04,113 squared. And all we have to do now is square these out and add 63 00:05:04,113 --> 00:05:08,524 them together 2 squares for three squared, 9 four squared 16 64 00:05:08,524 --> 00:05:10,529 and 5 squared is 25. 65 00:05:11,120 --> 00:05:14,574 And we can add these very simply. Foreign 16 is 20. 66 00:05:15,110 --> 00:05:21,545 9 and 25 is 3420 and 34 is 54. 67 00:05:22,490 --> 00:05:23,998 What could be simpler? 68 00:05:24,630 --> 00:05:30,520 Look at these two written using the Sigma notation, just 69 00:05:30,520 --> 00:05:35,821 slightly different from the previous ones. Remember we look 70 00:05:35,821 --> 00:05:38,177 at all the terms. 71 00:05:38,800 --> 00:05:43,672 Such as two to the K and we substitute in the values for K 72 00:05:43,672 --> 00:05:47,500 in increments of one starting with zero going up to five. 73 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 74 00:05:55,982 --> 00:06:03,434 we add 2 to the one +2 to the 2 + 2 to 3 + 2 to the 75 00:06:03,434 --> 00:06:05,918 4 + 2 to the five. 76 00:06:06,530 --> 00:06:10,202 Notice we've got six terms and the reason why we've got six 77 00:06:10,202 --> 00:06:13,568 terms is we started K at zero and not at one. 78 00:06:14,310 --> 00:06:21,707 And then we evaluate those two to the zero 1 + 2 plus 79 00:06:21,707 --> 00:06:29,104 force plus three 2 cubed, which is it +2 to 416. Let's do 80 00:06:29,104 --> 00:06:36,670 the 5:32. And when we add those up, 1 + 2 81 00:06:36,670 --> 00:06:41,220 is 347-1531. And then 63. 82 00:06:43,100 --> 00:06:45,850 Now look at this one. 83 00:06:46,100 --> 00:06:50,570 A little bit more involved, but quite straightforward if you 84 00:06:50,570 --> 00:06:55,934 follow the same steps that we've done before, we take the general 85 00:06:55,934 --> 00:07:00,851 term here and we substitute in values looking at where it 86 00:07:00,851 --> 00:07:06,662 starts from and where it ends. So R equals one is the lower 87 00:07:06,662 --> 00:07:12,473 limit, so we look at our general term and substituted are equal 1 88 00:07:12,473 --> 00:07:16,943 first and that will give us one times by two. 89 00:07:16,950 --> 00:07:20,846 Times by 1/2 added 90 00:07:20,846 --> 00:07:28,576 two. Now we substitute are equal to for the next one, so that's 91 00:07:28,576 --> 00:07:35,332 two are plus one is 3 + 1/2. An hour equals 3. 92 00:07:35,750 --> 00:07:38,040 3 plus One is 4. 93 00:07:38,590 --> 00:07:41,380 Plus 1/2 an. 94 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 95 00:07:48,839 --> 00:07:55,351 to it which is 5 and continue on in the same then until we get to 96 00:07:55,351 --> 00:08:01,049 our upper limit our equals 6. So that's a half six times by 7. 97 00:08:01,760 --> 00:08:05,881 And we can evaluate this side, but we can do a lot of 98 00:08:05,881 --> 00:08:09,685 canceling on the way before we do. And if the adding two 99 00:08:09,685 --> 00:08:13,489 cancers with the two here and here, 2 cancers with the four 100 00:08:13,489 --> 00:08:17,927 to give us two here 2 cancers with this forward to give us 2 101 00:08:17,927 --> 00:08:21,731 two counters with this six to give us three and two counters 102 00:08:21,731 --> 00:08:23,950 with this six to give us 3. 103 00:08:25,410 --> 00:08:31,052 And then we evaluate eyes adding together terms. So this one is 1 104 00:08:31,052 --> 00:08:38,278 plus. And we're left with 1 * 1 * 3, which is 3 105 00:08:38,278 --> 00:08:43,318 + 1 * 3 * 2, which is 6 plus. 106 00:08:44,220 --> 00:08:46,386 2 * 5 which is 10. 107 00:08:47,210 --> 00:08:52,677 Plus 3 * 5 which is 15 and the last 121. 108 00:08:53,250 --> 00:08:58,020 Now if you look at these numbers, these are quite 109 00:08:58,020 --> 00:09:02,313 special numbers. There's a name given to them, they 110 00:09:02,313 --> 00:09:03,744 either triangular numbers. 111 00:09:04,840 --> 00:09:08,788 And you should be able to recognize these the same way as 112 00:09:08,788 --> 00:09:11,749 you would recognize square numbers or prime numbers, or 113 00:09:11,749 --> 00:09:17,221 order even numbers. I'm finally what would like to do is just 114 00:09:17,221 --> 00:09:22,947 add these altogether. 1 + 3 + 6 is 10 + 10 is 20. 115 00:09:24,520 --> 00:09:31,060 20 + 1535 thirty six and then add the 20 is 56. 116 00:09:32,250 --> 00:09:34,506 So we get to some 56. 117 00:09:35,350 --> 00:09:38,020 Now, what 118 00:09:38,020 --> 00:09:44,436 about this? We've got our general term 2 to 119 00:09:44,436 --> 00:09:50,804 the K and we want to start with K equal 1 decay equal an if we 120 00:09:50,804 --> 00:09:55,580 write it out substituting in our values for K starting with K 121 00:09:55,580 --> 00:10:03,116 equal 1. Our first time will be 2 to the one plus then case two 122 00:10:03,116 --> 00:10:10,348 would be 2 to the 2 + 2 to 3 + 2 to the 4th and 123 00:10:10,348 --> 00:10:14,274 so on. Until we get to two to the N. 124 00:10:15,880 --> 00:10:19,354 Now we can't evaluate this to a finite number, 125 00:10:19,354 --> 00:10:23,214 because at this stage we don't know what Aaron is. 126 00:10:25,960 --> 00:10:33,144 And what if we have these two expressions? 127 00:10:33,990 --> 00:10:38,401 But still using the Sigma notation, but in this case are 128 00:10:38,401 --> 00:10:42,010 general term involves a negative sign in both cases. 129 00:10:42,610 --> 00:10:46,029 So we'll just take a little bit of time and carefully work the 130 00:10:46,029 --> 00:10:50,832 might. Because it's dead easy to go wrong in these sorts of ones, 131 00:10:50,832 --> 00:10:55,326 so take a little bit of time and you will hopefully work them out 132 00:10:55,326 --> 00:10:57,252 OK using the same steps that 133 00:10:57,252 --> 00:11:01,390 we've done before. In this particular example, our general 134 00:11:01,390 --> 00:11:07,620 term is negative, one raised to the part of our and R equals 1 135 00:11:07,620 --> 00:11:13,850 to 4. So the first term will be negative. One reason the part of 136 00:11:13,850 --> 00:11:19,773 one. Plus negative one race to the power of 2 plus negative 137 00:11:19,773 --> 00:11:24,646 one. Raise the pirate three and finally negative one raised to 138 00:11:24,646 --> 00:11:26,418 the power of 4. 139 00:11:27,260 --> 00:11:32,525 And then all we have to do is multiply these item, making sure 140 00:11:32,525 --> 00:11:37,385 that we get our signs correct, negative one raise the part of 141 00:11:37,385 --> 00:11:42,245 one is negative, one negative one raised to the part of two 142 00:11:42,245 --> 00:11:44,270 that's squared is positive one. 143 00:11:44,950 --> 00:11:47,038 Negative 1 cubed. 144 00:11:47,640 --> 00:11:52,008 That will give negative one and negative one is part of four. 145 00:11:52,008 --> 00:11:53,464 That'll give positive one. 146 00:11:54,560 --> 00:11:56,549 So in fact. 147 00:11:57,470 --> 00:12:02,891 Are some work site to be O negative 1 + 1 zero plus 148 00:12:02,891 --> 00:12:04,976 negative 1 + 1 zero? 149 00:12:05,700 --> 00:12:07,206 Quite a surprise, in fact, in 150 00:12:07,206 --> 00:12:10,790 fact. Now look at this one. 151 00:12:11,440 --> 00:12:16,632 Again, take care. Our general term is negative one over K 152 00:12:16,632 --> 00:12:22,768 squared and we have to use values of K from one to three. 153 00:12:23,650 --> 00:12:31,534 So we start with K equal 1. That's negative one over 1 154 00:12:31,534 --> 00:12:39,418 squared plus negative one over 2 squared plus negative one over 3 155 00:12:39,418 --> 00:12:45,659 squared. And then again, taking care with the negative signs. 156 00:12:45,659 --> 00:12:51,610 Square these values out negative one over one is negative, 1 157 00:12:51,610 --> 00:12:54,856 squared will give us one plus. 158 00:12:55,470 --> 00:13:00,606 Negative one over 2 all squared that would be positive and is 159 00:13:00,606 --> 00:13:03,174 1/2 * 1/2, which is quarter. 160 00:13:03,940 --> 00:13:07,603 Then added two negative one over 3 squared. Again, that's going 161 00:13:07,603 --> 00:13:12,598 to be positive, and it's going to be one over 3 * 1 over 3, 162 00:13:12,598 --> 00:13:14,929 which is one of her and I. 163 00:13:15,690 --> 00:13:20,019 We can add these together, adding the two fractions 164 00:13:20,019 --> 00:13:20,500 together. 165 00:13:21,560 --> 00:13:23,768 The common denominators 36. 166 00:13:24,310 --> 00:13:31,465 And it will be 9 + 4 and you've got your one there. So my 167 00:13:31,465 --> 00:13:37,666 final answer is one and 13 over 36. Quite a grotty answer, but. 168 00:13:38,180 --> 00:13:42,860 We've been able to workout what seemed to be quite a complicated 169 00:13:42,860 --> 00:13:44,030 Sigma notation some. 170 00:13:44,660 --> 00:13:46,646 And we finally got an answer. 171 00:13:47,660 --> 00:13:53,344 But what if we had along some and we wanted to write it in 172 00:13:53,344 --> 00:13:55,856 Sigma Notation? How would we do 173 00:13:55,856 --> 00:14:01,768 it? Well, why don't we first look at the two sums that we 174 00:14:01,768 --> 00:14:03,010 started off with? 175 00:14:03,020 --> 00:14:10,050 The first one was 1 + 2 + 3 + 176 00:14:10,050 --> 00:14:12,159 4 + 5. 177 00:14:12,930 --> 00:14:17,130 This one is quite an easy one to start off with because it's the 178 00:14:17,130 --> 00:14:21,147 sum. What type of terms? Well, it's a dead easy term because 179 00:14:21,147 --> 00:14:25,528 they are adding one on each time. So if we take the general 180 00:14:25,528 --> 00:14:26,539 term to BK. 181 00:14:28,040 --> 00:14:34,449 K starts the lower limit one and ends with the upper limit 5. 182 00:14:35,750 --> 00:14:39,240 So there's no problem with this one. This is very 183 00:14:39,240 --> 00:14:41,884 straightforward. But look at the 184 00:14:41,884 --> 00:14:46,720 second one. Again, it's not too bad because we notice and we 185 00:14:46,720 --> 00:14:48,650 said before that these are 186 00:14:48,650 --> 00:14:53,972 square numbers. So we could actually write these 187 00:14:53,972 --> 00:15:00,642 as one squared +2 squared, +3 squared, +4 squared, 5 188 00:15:00,642 --> 00:15:03,310 squared, and six squared. 189 00:15:04,740 --> 00:15:08,046 Once we've done this, are Sigma 190 00:15:08,046 --> 00:15:15,377 notation some? Is dead easy because we could easily see our 191 00:15:15,377 --> 00:15:18,935 general term must be K squared. 192 00:15:20,450 --> 00:15:23,411 And K must start with cake, will 193 00:15:23,411 --> 00:15:26,628 one. Takei equals 6. 194 00:15:27,240 --> 00:15:32,784 But what about these 195 00:15:32,784 --> 00:15:39,920 ones? This is a long some. 196 00:15:41,370 --> 00:15:46,290 What's different about this? It's got fractions and it also 197 00:15:46,290 --> 00:15:47,766 has alternating signs. 198 00:15:48,910 --> 00:15:52,995 Negative. Positive negative positive negative positive 199 00:15:52,995 --> 00:15:56,005 silver. Now the trick in this 200 00:15:56,005 --> 00:16:01,630 case. Is to rewrite all of these in terms of fractions. 201 00:16:02,740 --> 00:16:06,044 And then use negative 1 to a 202 00:16:06,044 --> 00:16:12,194 power. So if we look at this, we can rewrite this as negative one 203 00:16:12,194 --> 00:16:15,150 over 1. Plus 1/2. 204 00:16:16,830 --> 00:16:20,590 Minus one over 3 + 205 00:16:20,590 --> 00:16:25,459 1/4 plus. And So what to one over 100? 206 00:16:27,470 --> 00:16:33,302 Now we have to deal with the signs alternating. 207 00:16:34,620 --> 00:16:38,734 And if we think back we did have signs alternative before. 208 00:16:39,290 --> 00:16:42,782 And that's to do with negative numbers. Now we don't want to 209 00:16:42,782 --> 00:16:45,983 change the value of the fraction, we just want to change 210 00:16:45,983 --> 00:16:51,934 the sign. So if I rewrite this as negative one. 211 00:16:52,570 --> 00:16:54,720 Times by one over 1. 212 00:16:55,760 --> 00:16:59,710 Plus Negative one. 213 00:16:59,710 --> 00:17:01,398 The power of 2. 214 00:17:02,750 --> 00:17:04,019 One over 2. 215 00:17:05,010 --> 00:17:10,246 Plus negative 1 to the power of 3 one over 3. 216 00:17:10,780 --> 00:17:15,620 Plus negative 1 to the power of 4 one over 4. 217 00:17:17,230 --> 00:17:23,500 Time 2 plus negative one to the 101 over 100. 218 00:17:24,030 --> 00:17:25,968 Have we got the same sum? 219 00:17:26,890 --> 00:17:29,669 One yes, because look at each term. 220 00:17:31,760 --> 00:17:34,807 Negative 1 * 1 over one is negative, one over 1. 221 00:17:36,070 --> 00:17:40,965 Negative 1 squared will be positive 1 * 1/2, which will 222 00:17:40,965 --> 00:17:43,190 give us our plus 1/2. 223 00:17:43,850 --> 00:17:48,734 Negative 1 cubed will be negative one times by 1/3 which 224 00:17:48,734 --> 00:17:54,062 is negative 1/3, so that's where we get the negative 1/3 negative 225 00:17:54,062 --> 00:17:56,726 1 to the power of 4. 226 00:17:57,340 --> 00:18:01,825 Is positive one times by 1/4 gives us our plus one over 4. 227 00:18:02,940 --> 00:18:06,570 So rewriting our sequence that we started off with here. 228 00:18:07,360 --> 00:18:08,809 Into this format. 229 00:18:09,620 --> 00:18:13,820 That then helps us to write it in Sigma Notation. 230 00:18:15,430 --> 00:18:19,705 Because we can easily see that this is Sigma. 231 00:18:21,120 --> 00:18:24,950 We know it's negative one. 232 00:18:25,180 --> 00:18:31,879 To the power and the power is always the number underneath 233 00:18:31,879 --> 00:18:37,570 the fraction. So we'll say that our fraction is one over K 234 00:18:37,570 --> 00:18:42,764 because we start with K equal 1 and we raise negative 1 to the 235 00:18:42,764 --> 00:18:43,877 power of K. 236 00:18:44,570 --> 00:18:49,180 And to complete our Sigma notation for this long some. 237 00:18:50,050 --> 00:18:53,600 We start with the lower limit of K equal 1. 238 00:18:54,130 --> 00:18:57,105 And the upper limit K equal 100. 239 00:18:57,930 --> 00:19:00,926 So you can easily see that this 240 00:19:00,926 --> 00:19:03,429 notation. Is very concise. 241 00:19:04,100 --> 00:19:08,434 And I really lovely way of writing that big long some. 242 00:19:09,600 --> 00:19:16,950 Now I'd like to move on some special cases using 243 00:19:16,950 --> 00:19:21,744 Sigma notation. The first special case is when you have to 244 00:19:21,744 --> 00:19:24,290 sum a constant. Like this? 245 00:19:24,840 --> 00:19:29,351 If we have Sigma of three from K equal 1 to 5, what 246 00:19:29,351 --> 00:19:30,739 does that actually mean? 247 00:19:31,950 --> 00:19:33,940 Well, we work at night. 248 00:19:34,460 --> 00:19:41,740 We take each term for cake with one up to K equal 5. 249 00:19:42,660 --> 00:19:49,764 Each time remains the same. The matter whether cake was one K 250 00:19:49,764 --> 00:19:56,868 equals 2 cakes, 3, four, or five. So we actually just get 251 00:19:56,868 --> 00:19:59,828 the sum of 5 threes. 252 00:20:02,430 --> 00:20:04,380 Which is 15th. 253 00:20:06,080 --> 00:20:12,262 So with five terms, five times the constant to make 15. 254 00:20:13,170 --> 00:20:18,994 So we can get a general result from that if instead of three we 255 00:20:18,994 --> 00:20:20,658 have a constant C. 256 00:20:22,290 --> 00:20:26,520 And we take our K from one to N. 257 00:20:27,690 --> 00:20:32,790 What does this work? I to be? Well, we do it in the same way. 258 00:20:32,800 --> 00:20:35,062 We workout are some using cake 259 00:20:35,062 --> 00:20:39,580 with one. 2, three, and so on up to K equal an. 260 00:20:40,280 --> 00:20:41,600 And evaluate it. 261 00:20:42,100 --> 00:20:48,832 So can equal 1 is say K Equal 2C and so on. 262 00:20:50,780 --> 00:20:57,854 Up to. The last see such that we have and terms. 263 00:20:58,780 --> 00:21:05,176 Because we've got N lots of see this, some can be simplified 264 00:21:05,176 --> 00:21:08,374 dead easily to an times C. 265 00:21:10,620 --> 00:21:15,360 So this is a general result which is very useful when using 266 00:21:15,360 --> 00:21:18,915 Sigma notation. If you have to sum a constant. 267 00:21:19,700 --> 00:21:26,434 From cable one to N, then the value is and it is a finite 268 00:21:26,434 --> 00:21:28,839 value and times by C. 269 00:21:29,420 --> 00:21:30,988 What could be easier? 270 00:21:31,820 --> 00:21:34,668 But what about this? 271 00:21:35,910 --> 00:21:42,950 What if you had to sum a variable like this 3K? 272 00:21:43,480 --> 00:21:45,688 Can this be written more easily? 273 00:21:46,300 --> 00:21:48,390 Well, we'll take this particular 274 00:21:48,390 --> 00:21:53,924 example. 3K some 4K. Will want to four. 275 00:21:54,540 --> 00:22:01,400 So will sub in cake. Will 1 first, then K equal 2 then K 276 00:22:01,400 --> 00:22:07,280 equals 3 and then K equals 4 and we get that some. 277 00:22:07,820 --> 00:22:12,318 We notice that three is a factor if each one of these terms. 278 00:22:12,850 --> 00:22:20,172 So we factorize IR 3 and we get three bracket 1 + 2 + 279 00:22:20,172 --> 00:22:21,741 3 + 4. 280 00:22:23,220 --> 00:22:29,688 Now that is equal to three times by. If you want to add these 281 00:22:29,688 --> 00:22:34,308 altogether, that's three 610. So the answer is actually 30. 282 00:22:35,070 --> 00:22:39,102 Now I'm not so much interested in the answer, but in the 283 00:22:39,102 --> 00:22:43,990 process. At anyone stage, can you spot something that we know 284 00:22:43,990 --> 00:22:49,110 already? Well, look at this second line. You've got 3 times 285 00:22:49,110 --> 00:22:52,454 by 1 + 2 + 3 + 4. 286 00:22:53,770 --> 00:23:00,166 That's the addition of K&K is one 2, three and four, and we 287 00:23:00,166 --> 00:23:01,642 can rewrite this. 288 00:23:02,410 --> 00:23:06,174 Using Sigma notation that 289 00:23:06,174 --> 00:23:08,056 is 3. 290 00:23:08,580 --> 00:23:15,460 Sigma. Values K from K equals 291 00:23:15,460 --> 00:23:16,270 124. 292 00:23:17,500 --> 00:23:23,429 So our expression that we started off with Sigma of three 293 00:23:23,429 --> 00:23:29,897 K equals 1 to 4K was one to four equals 3 times. 294 00:23:30,470 --> 00:23:36,482 Sigma of K from cable one to four. So effectively we've taken 295 00:23:36,482 --> 00:23:41,492 out with factorized at three from our first expression, and 296 00:23:41,492 --> 00:23:44,498 we brought it outside the Sigma 297 00:23:44,498 --> 00:23:48,200 notation. So more generally. 298 00:23:48,820 --> 00:23:56,506 We can take that if we've got the Sigma of CK from K equals 299 00:23:56,506 --> 00:23:58,153 1 to N. 300 00:23:58,940 --> 00:24:00,160 What would that equal? 301 00:24:00,680 --> 00:24:07,610 Wow, quite easy. All we have to do is that we see that that is. 302 00:24:08,280 --> 00:24:14,869 See times one plus C times 2 plus C times 3. 303 00:24:16,020 --> 00:24:17,109 Right up to. 304 00:24:17,680 --> 00:24:20,571 See times an. We've got M terms 305 00:24:20,571 --> 00:24:23,665 here. We can factorize by the 306 00:24:23,665 --> 00:24:30,980 sea. And we get 1 + 2 + 3 plus up 307 00:24:30,980 --> 00:24:37,190 to N and then we can rewrite this using Sigma 308 00:24:37,190 --> 00:24:44,021 notation, that is C Sigma of care from K equals 1 309 00:24:44,021 --> 00:24:45,263 to N. 310 00:24:46,440 --> 00:24:47,548 And as I say. 311 00:24:48,420 --> 00:24:53,337 Quite straightforward, when you've got a Sigma of C times a 312 00:24:53,337 --> 00:24:57,807 variable. See being a constant, you can take the constant 313 00:24:57,807 --> 00:25:03,171 outside the Sigma notation and you left with C times by the 314 00:25:03,171 --> 00:25:06,747 Sigma of K from K equal 1 to 315 00:25:06,747 --> 00:25:11,733 N. That's another very useful result that you should remember 316 00:25:11,733 --> 00:25:13,401 when using Sigma notation. 317 00:25:14,190 --> 00:25:20,185 But what if we had this expression? The sum of the 318 00:25:20,185 --> 00:25:26,725 variable K plus two where K takes the lower limit one to 319 00:25:26,725 --> 00:25:33,810 four? What does that work? I'd be? Can we write it more concert 320 00:25:33,810 --> 00:25:35,990 concisely and easily well? 321 00:25:36,690 --> 00:25:40,988 We do it stage by stage, step by step, putting in values for K 322 00:25:40,988 --> 00:25:42,830 from K equal 1 to 4. 323 00:25:43,480 --> 00:25:47,310 So the first time case one says 1 + 2. 324 00:25:48,720 --> 00:25:53,780 Plus second, her K equals 2, so that's 2 + 2. 325 00:25:54,470 --> 00:26:01,370 The third term is 3 + 2 and the fourth term is 4 + 2. 326 00:26:02,580 --> 00:26:07,926 Now what I want to do next is rearrange these values. 327 00:26:09,040 --> 00:26:15,823 If I rearrange such that I have the addition of 1 + 2 + 3 + 4 328 00:26:15,823 --> 00:26:23,114 first. And then see what we've got left. We've got 329 00:26:23,114 --> 00:26:29,974 add 2 four times, so that is 4 * 2. 330 00:26:31,420 --> 00:26:37,630 And then we can easily evaluate this expression 1 + 2 + 3 + 4 331 00:26:37,630 --> 00:26:41,356 is 10 + 4 two 38. So the answer 332 00:26:41,356 --> 00:26:46,898 is 18. But as I said before, we're not so much concerned with 333 00:26:46,898 --> 00:26:51,910 the answer, but in the process, can we look and see at any line? 334 00:26:52,590 --> 00:26:54,948 Can we use the Sigma notation 335 00:26:54,948 --> 00:26:58,050 more easily? Well, if we look at 336 00:26:58,050 --> 00:27:03,686 this line. We've seen part of this before. 337 00:27:04,370 --> 00:27:07,506 1 + 2 + 3 + 4. 338 00:27:08,640 --> 00:27:14,305 Can be written more easily using Sigma notation as Sigma. Care 339 00:27:14,305 --> 00:27:17,395 from K equals 1 to 4. 340 00:27:18,080 --> 00:27:23,337 Plus then we've got 4 * 2. 341 00:27:25,430 --> 00:27:31,022 So looking back at what we started with, we've got Sigma of 342 00:27:31,022 --> 00:27:38,012 K plus two from K equals one to four and it expanded I to this 343 00:27:38,012 --> 00:27:44,536 Sigma notation Sigma of can at all, from cable one to 4 + 4 344 00:27:44,536 --> 00:27:50,594 * 2 or two is the constant that was added to the care. 345 00:27:52,410 --> 00:27:54,828 So we can generalize this result. 346 00:27:56,950 --> 00:28:01,290 If we had Sigma of. 347 00:28:02,340 --> 00:28:08,820 G of K function of K plus a constant C. 348 00:28:09,320 --> 00:28:16,509 As our variable and we want to take care from one to N. 349 00:28:17,910 --> 00:28:23,430 We can use the pattern that we spotted here to write this 350 00:28:23,430 --> 00:28:27,110 out using Sigma notation plus some other value. 351 00:28:28,410 --> 00:28:32,070 The first part is Sigma. 352 00:28:32,760 --> 00:28:38,545 Off the function that we started off with in our variable and the 353 00:28:38,545 --> 00:28:40,325 function is GF care. 354 00:28:41,010 --> 00:28:48,270 And we see that it should be from K to the upper limit from K 355 00:28:48,270 --> 00:28:54,548 equal 1. Takei equal the upper limit in this case is for, but 356 00:28:54,548 --> 00:28:57,334 in our general case it's K equal 357 00:28:57,334 --> 00:29:00,720 N. Says K equal 1 to an. 358 00:29:01,320 --> 00:29:08,384 Plus Now the next part, remember it was the upper limit by the 359 00:29:08,384 --> 00:29:13,688 number of terms times by the constant that you've had in your 360 00:29:13,688 --> 00:29:15,898 variable. Well, look here. Are 361 00:29:15,898 --> 00:29:22,748 constantly see. And our upper variable is an. 362 00:29:23,280 --> 00:29:25,870 So it's End Times C. 363 00:29:26,780 --> 00:29:30,255 So this Sigma notation can 364 00:29:30,255 --> 00:29:37,540 be expanded. Two Sigma GF K from K Equal 1M Plus M 365 00:29:37,540 --> 00:29:38,590 Times C. 366 00:29:40,210 --> 00:29:44,746 And we can even generalize further using previous results 367 00:29:44,746 --> 00:29:49,786 if we took Sigma from K equals 1 to N. 368 00:29:50,370 --> 00:29:54,210 Of a GF K 369 00:29:54,210 --> 00:29:57,820 Plus C. As our 370 00:29:57,820 --> 00:30:01,368 variable. We can expand that 371 00:30:01,368 --> 00:30:07,750 out. Looking at this time 1st and then dealing with the 372 00:30:07,750 --> 00:30:12,997 constant now, this term is a multiple a constant times by 373 00:30:12,997 --> 00:30:18,721 your GF K so we can bring that outside the Sigma notation. 374 00:30:18,810 --> 00:30:26,766 And say it's a Times by Sigma GfK from K equals 1 375 00:30:26,766 --> 00:30:28,092 to N. 376 00:30:29,560 --> 00:30:30,630 Plus 377 00:30:32,670 --> 00:30:38,065 Just what's left to do with the sea and we know when that's 378 00:30:38,065 --> 00:30:41,385 inside the bracket when we expand it out. 379 00:30:42,070 --> 00:30:44,416 We're left with end times C. 380 00:30:45,730 --> 00:30:46,810 What could be easier? 381 00:30:47,560 --> 00:30:51,260 All right? But 382 00:30:51,260 --> 00:30:57,566 what if? We wanted to use Sigma notation and the 383 00:30:57,566 --> 00:31:03,350 variable we wanted to add together is K Plus K squared IE 384 00:31:03,350 --> 00:31:05,278 two functions of K. 385 00:31:07,060 --> 00:31:11,053 What would we get when we write out this long some? 386 00:31:11,660 --> 00:31:14,480 Start simply by substituting in 387 00:31:14,480 --> 00:31:20,683 for K. Case one to begin with. So the first term is 1 + 1 388 00:31:20,683 --> 00:31:28,043 squared. Second time is 2 + 2 squared. Third time is 3 + 3 389 00:31:28,043 --> 00:31:33,224 squared, and that's our final term because the upper limit for 390 00:31:33,224 --> 00:31:34,637 K is 3. 391 00:31:35,300 --> 00:31:41,528 Now we'll rearrange the way we did before to make our work 392 00:31:41,528 --> 00:31:46,718 easier. Will take the 1 + 2 + 3 together. 393 00:31:49,040 --> 00:31:53,473 And then we'll take the one squared and the two squared 394 00:31:53,473 --> 00:31:56,697 and the three squared bits that are left. 395 00:31:57,860 --> 00:32:02,684 And you can see why I've done that, because here we're adding 396 00:32:02,684 --> 00:32:05,096 our constant case and here we're 397 00:32:05,096 --> 00:32:07,828 adding. Our case squareds. 398 00:32:08,750 --> 00:32:14,282 When we add these together, we just add them up and multiply 399 00:32:14,282 --> 00:32:20,736 out using the square sign you get 1 + 2 + 3 is 6, 400 00:32:20,736 --> 00:32:26,729 one squared is 1, two squared is 4 three squared 9. Add back 401 00:32:26,729 --> 00:32:31,339 together that gives you fourteen. 6 + 14 is 20. 402 00:32:33,160 --> 00:32:36,286 But look at this second line. 403 00:32:37,420 --> 00:32:42,685 We can simplify this second line using Sigma notation. 404 00:32:43,510 --> 00:32:48,762 And it's using Sigma notation that we've done before. 1 + 2 + 405 00:32:48,762 --> 00:32:54,418 3 is the same as Sigma of care from K equals 1 to 3. 406 00:32:55,350 --> 00:32:56,100 Plus 407 00:32:57,480 --> 00:33:02,560 1 squared, +2 squared, +3 squared, that's dead easy. It's 408 00:33:02,560 --> 00:33:09,164 the sum of all variables, K squared from K equals 1 to 3. 409 00:33:10,320 --> 00:33:14,159 Now look at what would started and look what we've got. 410 00:33:14,870 --> 00:33:21,580 We have broken this Sigma notation up into two sums to 411 00:33:21,580 --> 00:33:28,540 Sigma sums. So here we've got the Sigma offer variable 412 00:33:28,540 --> 00:33:30,490 plus another variable. 413 00:33:31,420 --> 00:33:34,000 And with split it up to. 414 00:33:34,720 --> 00:33:39,364 Sigma of one variable added to the Sigma of the other variable. 415 00:33:40,420 --> 00:33:42,330 We've used the distributive law. 416 00:33:43,350 --> 00:33:46,880 So to write this more 417 00:33:46,880 --> 00:33:52,133 generally. We can say that the sum of a function of care. 418 00:33:53,140 --> 00:33:57,466 Added to another function of K. 419 00:33:58,950 --> 00:34:04,610 From K equals 1 to N must equal the sum. 420 00:34:05,900 --> 00:34:13,284 Two separate sums. The sum of JFK from K equals 1 to M 421 00:34:13,284 --> 00:34:19,755 Plus Sigma. FFK from K equals 1 to add. 422 00:34:21,580 --> 00:34:26,546 And in fact, we can continue this on if we added on other 423 00:34:26,546 --> 00:34:28,074 functions within our variable 424 00:34:28,074 --> 00:34:34,606 here. Then all we have to do is add on another Sigma of 425 00:34:34,606 --> 00:34:35,965 that particular function. 426 00:34:37,980 --> 00:34:43,047 Finally, here's a real life example using Sigma notation. 427 00:34:46,440 --> 00:34:50,388 Take for example if we want to find the mean of a set of 428 00:34:50,388 --> 00:34:52,926 numbers. That could be the marks in a test. 429 00:34:54,230 --> 00:34:58,790 If we want to find the main what we want to do is we have to 430 00:34:58,790 --> 00:34:59,930 workout the total sum. 431 00:35:01,500 --> 00:35:04,530 Divided by the number of values. 432 00:35:05,580 --> 00:35:10,816 And that's what the main is. Well, that's actually what we've 433 00:35:10,816 --> 00:35:15,576 been doing in the previous examples. We have been finding 434 00:35:15,576 --> 00:35:21,764 the total sum, and we have been looking at the number of values. 435 00:35:21,764 --> 00:35:27,952 So if I take this example, just say we had the marks 23456. 436 00:35:29,210 --> 00:35:34,150 And we want to find the mean of those marks. The main will 437 00:35:34,150 --> 00:35:41,924 equal. 2 + 3 + 4 + 5 + 6 and we'd have to divide 438 00:35:41,924 --> 00:35:48,112 it by the number of values, which is 5 when we work that 439 00:35:48,112 --> 00:35:51,444 out. That is 5 + 510, another 440 00:35:51,444 --> 00:35:57,056 10. 20 over 5 which works out to be the value 4. 441 00:35:57,910 --> 00:35:59,650 But more generally. 442 00:36:00,210 --> 00:36:07,280 If you've got a set of marks, say XI, we can write the main 443 00:36:07,280 --> 00:36:13,845 in terms of Sigma Notation. It's the total sum of all the marks. 444 00:36:14,380 --> 00:36:15,829 Such as Zhao. 445 00:36:17,630 --> 00:36:20,090 From I equals 1 to N. 446 00:36:20,830 --> 00:36:25,940 And then we want to divide it by the number of values. Now we 447 00:36:25,940 --> 00:36:27,765 know there are N Marks. 448 00:36:28,570 --> 00:36:35,738 So we want to pie by one over N, so the main is equal to one 449 00:36:35,738 --> 00:36:41,114 over an Sigma X of I from I equal 1 to M.