I was playing around with my calculator during class the other day. (Yes, I still don't pay attention in class.) In doing so, I discovered something.

 

[math]n^2=1+2+3+...+(n-1)+n+(n-1)+...+3+2+1[/math]

 

Is there a name for this?

This is fairly basic math.

 

In fact, if you sum up powers of i, with i ranging from 1 to n, then you'll see that summing a k-th power yields a number of k+1th power (plus some correction terms of lower power).

 

This can be connected to integrating functions. If you take the integral of x^k, then the result has power k+1. In your case, you can connect it to integrating x, which gives an expression of order x^2.

So, no name?

I'm thinking perhaps "factorial"?

 

i.e. 5! = 5x4x3x2x1

 

n(n-1)(n-2)(n-3)...1

 

But it probably doesn't fit with your pattern because you are adding the terms. It looks like if a person is starting at the bottom of a mountain, and it's going uphill. Then at the top of the peak, it goes downhill back to the bottom.

 

Anyways.

It would usually be done wth mathematical induction - each time, you add 2n + 1, making the numbe ron the left side n^2 + 2n + 1 = (n+1)^2, and as you have a base case with 1 working, mathematical induction guarantees it for all n.

=Uncool-

Arithmetic Series.

 

Get the value of [math]1+2+3+...+n[/math], add it to [math](n-1)+(n-2)+...+1[/math]:

 

[math]\frac{n(n+1)}{2}+\frac{n(n-1)}{2}[/math]

 

Simplify, gives the series as [math]n^2[/math].

Or, if you prefer a geometric equivalent, the 2 halves of the expression on the right are 2 triangles, put together with a line of n "stars" they form a square.

* ***

** **

*** *

****

Can someone who understood that and who has a better graphics package than me draw that please?

Hey, that's quite nice!

 

Put the series in pairs like this:

 

[math]1+n-1+2+n-2+...+n-1+1+n[/math]

[math]=n+n+n...+n[/math]. There are [math]n[/math] n's adding together. Which is the square that John drew above.

So: [math]n^2[/math] is the series's sum.

Hey, that's quite nice!

 

Put the series in pairs like this:

 

[math]1+n-1+2+n-2+...+n-1+1+n[/math]

[math]=n+n+n...+n[/math]. There are [math]n[/math] n's adding together. Which is the square that John drew above.

So: [math]n^2[/math] is the series's sum.

 

Very nice and quite emphatic.

cool