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Let's do another summation problem. The summation problem that I want to do,

Â is the sum as n goes from 1 to a some number k of just n.

Â Alright? In other words, I want to add up 1 plus 2

Â plus 3 dot, dot, dot plus k, right. I want to add up the first k whole

Â numbers. These are called triangular numbers.

Â The reason is because performing this sum is really counting the dots in a triangle.

Â Let me draw a picture. So, 1 dot, 2 dots Three dots, four dots,

Â five dots. All right?

Â And then I'll keep on going until I get to K dots.

Â You know, and this picture here is a triangle.

Â Right? So this sum is counting the number of dots

Â in a triangle. Having that k there makes this a very

Â general problem, and anytime that you're confronted with a general problem, it

Â helps to specialize to a specific case. In order to gain some insight.

Â So let's set k equal to 10. So I'm just going to do the sum n goes

Â from 1 to k which is now 10 of n. Which means I'm going to add 1 plus 2 plus

Â 3 plus 4 plus 5 plus 6 plus 7 plus 8 plus 9 plus 10.

Â I could actually draw this as a triangle. Put down 1 dot, 2 dots, 3 dots, 4 dots, 5

Â dots, 6 dots, 7 dots, 8 dots, 9 dots, 10 dots.

Â [laugh] So I've got this triangle, and this sum is just the number of dots in

Â this triangle. Now I just count those dots.

Â I just add those numbers up. 1 plus 2 is 3, 3 plus 3 is 6, 6 plus 4 is

Â 10, 10 plus 5 is 15, 15 plus 6 is 21, 21 plus 7 is 28, 28 plus 8 is 36, 36 plus 9

Â is 45, and 45 plus 10. Is 55.

Â So the sum of the first 10 whole numbers is 55, and I drew 55 dots to make this

Â triangle. But there's another approach.

Â Instead of just adding 1 plus 2 plus 3 plus 4 plus 5 plus 6 plus 7 plus 8 plus 9

Â plus 10, I could add these in a different order.

Â I could add 1 and 10 together to make 11. 2 and 9 together to make 11.

Â 3 and 8 together to make 11. 4 and 7 together to make 11 , and I'm left

Â with 5 and 6 and those two together make 11, right?

Â So I've got five groups of 11, which means if I add up all these numbers I get 55.

Â Armed with this trick I can do the same thing to attack the problem in general.

Â So in general, I'm trying to do the sum, n goes from 1 to k of just n.

Â Trying to add up the first k whole numbers.

Â And if I write this in a different order, and let's say k is even, just to make this

Â a little bit easier to think about. Alright I'll take the first number and the

Â last number, and add those together. The second number and the second-to-last

Â number and add those together. The third number and the third-to-last

Â number, and add those together. And I'm going to keep on going, right?

Â And the first number and the last number is, you know, I'm writing it as 1 plus K,

Â that's fine. The second number and the second-to-last

Â number, well that's also 1 plus K. All right, the third number and the

Â third-to-last that's also 1 plus K. Right?

Â All of these things add up to 1 plus K. And the key question is just how many

Â things I've got. All right?

Â And I'm pairing them off in twos. So, let's say this is an even number.

Â K is an even number. And I've got K over 2 pairs.

Â All of which add up to K plus 1. And that's just a multiplication problem.

Â Right? That means that this sum is 1 plus k times

Â k over 2. So that's the sum of the first k whole

Â numbers. If all this algebra doesn't really speak

Â to you. We can redo this geometrically.

Â So what I want to calculate, right, is this sum.

Â The sum as n goes from 1 to k of n. And geometrically, right, I can draw this

Â pattern of dots, right? 1 plus 2 plus 3 plus 4 plus 5 plus 6 in

Â this case. But I could imagine that, you know, this

Â is a specific picture that represents the general case, right?

Â I've got this triangle, a k by k triangle of dots.

Â And this sum is just counting the number of dots.

Â Right? 1 plus 2 plus 3, that's exactly what the

Â sum does. Now I'm going to count these dots.

Â So I'll make two copies of this picture. So now I've got two copies of this sum.

Â Right? There's another k by k triangle of dots.

Â And to count these dots I'm going to take this triangle, rotate it, and slide it

Â over here. So as to make not a k by k square, but

Â this is a k plus 1 by k rectangle, alright?

Â As the bottom of this one just fits in, but it's a little bit wider.

Â I got this k plus 1 by k rectangle of dots.

Â So I know there's k plus 1 times k dots and this is two copies of the sum that I'm

Â interested in. So 2 times the sum I'm interested in is k

Â plus 1 times k, the number of dots in this rectangle.

Â And consequently if I just divide by 2 I get the formula that I'm interested in.

Â The sum of n, n goes from 1 to k is k plus 1 times k over 2, right.

Â It's half of the number of dots in this k plus 1 by k rectangle.

Â A big part of the joy of mathematics is that the same ideas can be presented with

Â very different clothing. In this case, we're seeing really the same

Â argument. But presented both algebraically and

Â geometrically.

Â