Let's do another summation problem. The summation problem that I want to do,
is the sum as n goes from 1 to a some number k of just n.
Alright? In other words, I want to add up 1 plus 2
plus 3 dot, dot, dot plus k, right. I want to add up the first k whole
numbers. These are called triangular numbers.
The reason is because performing this sum is really counting the dots in a triangle.
Let me draw a picture. So, 1 dot, 2 dots Three dots, four dots,
five dots. All right?
And then I'll keep on going until I get to K dots.
You know, and this picture here is a triangle.
Right? So this sum is counting the number of dots
in a triangle. Having that k there makes this a very
general problem, and anytime that you're confronted with a general problem, it
helps to specialize to a specific case. In order to gain some insight.
So let's set k equal to 10. So I'm just going to do the sum n goes
from 1 to k which is now 10 of n. Which means I'm going to add 1 plus 2 plus
3 plus 4 plus 5 plus 6 plus 7 plus 8 plus 9 plus 10.
I could actually draw this as a triangle. Put down 1 dot, 2 dots, 3 dots, 4 dots, 5
dots, 6 dots, 7 dots, 8 dots, 9 dots, 10 dots.
[laugh] So I've got this triangle, and this sum is just the number of dots in
this triangle. Now I just count those dots.
I just add those numbers up. 1 plus 2 is 3, 3 plus 3 is 6, 6 plus 4 is
10, 10 plus 5 is 15, 15 plus 6 is 21, 21 plus 7 is 28, 28 plus 8 is 36, 36 plus 9
is 45, and 45 plus 10. Is 55.
So the sum of the first 10 whole numbers is 55, and I drew 55 dots to make this
triangle. But there's another approach.
Instead of just adding 1 plus 2 plus 3 plus 4 plus 5 plus 6 plus 7 plus 8 plus 9
plus 10, I could add these in a different order.
I could add 1 and 10 together to make 11. 2 and 9 together to make 11.
3 and 8 together to make 11. 4 and 7 together to make 11 , and I'm left
with 5 and 6 and those two together make 11, right?
So I've got five groups of 11, which means if I add up all these numbers I get 55.
Armed with this trick I can do the same thing to attack the problem in general.
So in general, I'm trying to do the sum, n goes from 1 to k of just n.
Trying to add up the first k whole numbers.
And if I write this in a different order, and let's say k is even, just to make this
a little bit easier to think about. Alright I'll take the first number and the
last number, and add those together. The second number and the second-to-last
number and add those together. The third number and the third-to-last
number, and add those together. And I'm going to keep on going, right?
And the first number and the last number is, you know, I'm writing it as 1 plus K,
that's fine. The second number and the second-to-last
number, well that's also 1 plus K. All right, the third number and the
third-to-last that's also 1 plus K. Right?
All of these things add up to 1 plus K. And the key question is just how many
things I've got. All right?
And I'm pairing them off in twos. So, let's say this is an even number.
K is an even number. And I've got K over 2 pairs.
All of which add up to K plus 1. And that's just a multiplication problem.
Right? That means that this sum is 1 plus k times
k over 2. So that's the sum of the first k whole
numbers. If all this algebra doesn't really speak
to you. We can redo this geometrically.
So what I want to calculate, right, is this sum.
The sum as n goes from 1 to k of n. And geometrically, right, I can draw this
pattern of dots, right? 1 plus 2 plus 3 plus 4 plus 5 plus 6 in
this case. But I could imagine that, you know, this
is a specific picture that represents the general case, right?
I've got this triangle, a k by k triangle of dots.
And this sum is just counting the number of dots.
Right? 1 plus 2 plus 3, that's exactly what the
sum does. Now I'm going to count these dots.
So I'll make two copies of this picture. So now I've got two copies of this sum.
Right? There's another k by k triangle of dots.
And to count these dots I'm going to take this triangle, rotate it, and slide it
over here. So as to make not a k by k square, but
this is a k plus 1 by k rectangle, alright?
As the bottom of this one just fits in, but it's a little bit wider.
I got this k plus 1 by k rectangle of dots.
So I know there's k plus 1 times k dots and this is two copies of the sum that I'm
interested in. So 2 times the sum I'm interested in is k
plus 1 times k, the number of dots in this rectangle.
And consequently if I just divide by 2 I get the formula that I'm interested in.
The sum of n, n goes from 1 to k is k plus 1 times k over 2, right.
It's half of the number of dots in this k plus 1 by k rectangle.
A big part of the joy of mathematics is that the same ideas can be presented with
very different clothing. In this case, we're seeing really the same
argument. But presented both algebraically and
geometrically.