1 00:00:05,380 --> 00:00:10,586 Let's do another summation problem. The summation problem that I want to do, 2 00:00:10,586 --> 00:00:14,765 is the sum as n goes from 1 to a some number k of just n. 3 00:00:14,765 --> 00:00:19,681 Alright? In other words, I want to add up 1 plus 2 4 00:00:19,681 --> 00:00:25,840 plus 3 dot, dot, dot plus k, right. I want to add up the first k whole 5 00:00:25,840 --> 00:00:29,469 numbers. These are called triangular numbers. 6 00:00:29,469 --> 00:00:34,270 The reason is because performing this sum is really counting the dots in a triangle. 7 00:00:34,270 --> 00:00:42,282 Let me draw a picture. So, 1 dot, 2 dots Three dots, four dots, 8 00:00:42,282 --> 00:00:45,054 five dots. All right? 9 00:00:45,055 --> 00:00:49,597 And then I'll keep on going until I get to K dots. 10 00:00:49,598 --> 00:00:52,802 You know, and this picture here is a triangle. 11 00:00:52,802 --> 00:00:56,196 Right? So this sum is counting the number of dots 12 00:00:56,196 --> 00:01:00,290 in a triangle. Having that k there makes this a very 13 00:01:00,290 --> 00:01:05,583 general problem, and anytime that you're confronted with a general problem, it 14 00:01:05,583 --> 00:01:10,263 helps to specialize to a specific case. In order to gain some insight. 15 00:01:10,263 --> 00:01:15,545 So let's set k equal to 10. So I'm just going to do the sum n goes 16 00:01:15,545 --> 00:01:22,852 from 1 to k which is now 10 of n. Which means I'm going to add 1 plus 2 plus 17 00:01:22,852 --> 00:01:28,285 3 plus 4 plus 5 plus 6 plus 7 plus 8 plus 9 plus 10. 18 00:01:28,285 --> 00:01:39,569 I could actually draw this as a triangle. Put down 1 dot, 2 dots, 3 dots, 4 dots, 5 19 00:01:39,569 --> 00:01:49,290 dots, 6 dots, 7 dots, 8 dots, 9 dots, 10 dots. 20 00:01:49,291 --> 00:01:55,801 [laugh] So I've got this triangle, and this sum is just the number of dots in 21 00:01:55,801 --> 00:01:59,000 this triangle. Now I just count those dots. 22 00:01:59,000 --> 00:02:04,856 I just add those numbers up. 1 plus 2 is 3, 3 plus 3 is 6, 6 plus 4 is 23 00:02:04,856 --> 00:02:13,498 10, 10 plus 5 is 15, 15 plus 6 is 21, 21 plus 7 is 28, 28 plus 8 is 36, 36 plus 9 24 00:02:13,498 --> 00:02:17,996 is 45, and 45 plus 10. Is 55. 25 00:02:17,996 --> 00:02:23,368 So the sum of the first 10 whole numbers is 55, and I drew 55 dots to make this 26 00:02:23,368 --> 00:02:26,396 triangle. But there's another approach. 27 00:02:26,396 --> 00:02:31,076 Instead of just adding 1 plus 2 plus 3 plus 4 plus 5 plus 6 plus 7 plus 8 plus 9 28 00:02:31,076 --> 00:02:34,124 plus 10, I could add these in a different order. 29 00:02:34,124 --> 00:02:39,665 I could add 1 and 10 together to make 11. 2 and 9 together to make 11. 30 00:02:39,665 --> 00:02:44,684 3 and 8 together to make 11. 4 and 7 together to make 11 , and I'm left 31 00:02:44,684 --> 00:02:48,115 with 5 and 6 and those two together make 11, right? 32 00:02:48,115 --> 00:02:54,340 So I've got five groups of 11, which means if I add up all these numbers I get 55. 33 00:02:54,340 --> 00:02:59,742 Armed with this trick I can do the same thing to attack the problem in general. 34 00:02:59,743 --> 00:03:05,990 So in general, I'm trying to do the sum, n goes from 1 to k of just n. 35 00:03:05,990 --> 00:03:09,187 Trying to add up the first k whole numbers. 36 00:03:09,188 --> 00:03:12,830 And if I write this in a different order, and let's say k is even, just to make this 37 00:03:12,830 --> 00:03:16,875 a little bit easier to think about. Alright I'll take the first number and the 38 00:03:16,875 --> 00:03:22,460 last number, and add those together. The second number and the second-to-last 39 00:03:22,460 --> 00:03:27,842 number and add those together. The third number and the third-to-last 40 00:03:27,842 --> 00:03:32,004 number, and add those together. And I'm going to keep on going, right? 41 00:03:32,005 --> 00:03:36,755 And the first number and the last number is, you know, I'm writing it as 1 plus K, 42 00:03:36,755 --> 00:03:40,725 that's fine. The second number and the second-to-last 43 00:03:40,725 --> 00:03:44,618 number, well that's also 1 plus K. All right, the third number and the 44 00:03:44,618 --> 00:03:46,882 third-to-last that's also 1 plus K. Right? 45 00:03:46,882 --> 00:03:51,200 All of these things add up to 1 plus K. And the key question is just how many 46 00:03:51,200 --> 00:03:53,344 things I've got. All right? 47 00:03:53,344 --> 00:03:56,750 And I'm pairing them off in twos. So, let's say this is an even number. 48 00:03:56,750 --> 00:04:00,480 K is an even number. And I've got K over 2 pairs. 49 00:04:00,480 --> 00:04:05,182 All of which add up to K plus 1. And that's just a multiplication problem. 50 00:04:05,182 --> 00:04:10,663 Right? That means that this sum is 1 plus k times 51 00:04:10,663 --> 00:04:15,200 k over 2. So that's the sum of the first k whole 52 00:04:15,200 --> 00:04:18,388 numbers. If all this algebra doesn't really speak 53 00:04:18,388 --> 00:04:21,404 to you. We can redo this geometrically. 54 00:04:21,405 --> 00:04:23,750 So what I want to calculate, right, is this sum. 55 00:04:23,750 --> 00:04:29,740 The sum as n goes from 1 to k of n. And geometrically, right, I can draw this 56 00:04:29,740 --> 00:04:34,510 pattern of dots, right? 1 plus 2 plus 3 plus 4 plus 5 plus 6 in 57 00:04:34,510 --> 00:04:37,127 this case. But I could imagine that, you know, this 58 00:04:37,127 --> 00:04:39,944 is a specific picture that represents the general case, right? 59 00:04:39,945 --> 00:04:44,160 I've got this triangle, a k by k triangle of dots. 60 00:04:44,160 --> 00:04:47,471 And this sum is just counting the number of dots. 61 00:04:47,472 --> 00:04:49,875 Right? 1 plus 2 plus 3, that's exactly what the 62 00:04:49,875 --> 00:04:52,450 sum does. Now I'm going to count these dots. 63 00:04:52,450 --> 00:04:57,056 So I'll make two copies of this picture. So now I've got two copies of this sum. 64 00:04:57,056 --> 00:05:00,139 Right? There's another k by k triangle of dots. 65 00:05:00,140 --> 00:05:06,471 And to count these dots I'm going to take this triangle, rotate it, and slide it 66 00:05:06,471 --> 00:05:10,526 over here. So as to make not a k by k square, but 67 00:05:10,526 --> 00:05:14,393 this is a k plus 1 by k rectangle, alright? 68 00:05:14,394 --> 00:05:19,513 As the bottom of this one just fits in, but it's a little bit wider. 69 00:05:19,513 --> 00:05:22,754 I got this k plus 1 by k rectangle of dots. 70 00:05:22,754 --> 00:05:28,052 So I know there's k plus 1 times k dots and this is two copies of the sum that I'm 71 00:05:28,052 --> 00:05:31,652 interested in. So 2 times the sum I'm interested in is k 72 00:05:31,652 --> 00:05:34,964 plus 1 times k, the number of dots in this rectangle. 73 00:05:34,964 --> 00:05:40,200 And consequently if I just divide by 2 I get the formula that I'm interested in. 74 00:05:40,200 --> 00:05:44,627 The sum of n, n goes from 1 to k is k plus 1 times k over 2, right. 75 00:05:44,628 --> 00:05:49,690 It's half of the number of dots in this k plus 1 by k rectangle. 76 00:05:49,690 --> 00:05:54,955 A big part of the joy of mathematics is that the same ideas can be presented with 77 00:05:54,955 --> 00:05:58,865 very different clothing. In this case, we're seeing really the same 78 00:05:58,865 --> 00:06:07,610 argument. But presented both algebraically and 79 00:06:07,610 --> 00:06:11,188 geometrically.