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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.

$n^2=1+2+3+...+(n-1)+n+(n-1)+...+3+2+1$

Is there a name for this?

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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?

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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.

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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-

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Arithmetic Series.

Get the value of $1+2+3+...+n$, add it to $(n-1)+(n-2)+...+1$:

$\frac{n(n+1)}{2}+\frac{n(n-1)}{2}$

Simplify, gives the series as $n^2$.

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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?

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Hey, that's quite nice!

Put the series in pairs like this:

$1+n-1+2+n-2+...+n-1+1+n$

$=n+n+n...+n$. There are $n$ n's adding together. Which is the square that John drew above.

So: $n^2$ is the series's sum.

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• 1 month later...
Hey, that's quite nice!

Put the series in pairs like this:

$1+n-1+2+n-2+...+n-1+1+n$

$=n+n+n...+n$. There are $n$ n's adding together. Which is the square that John drew above.

So: $n^2$ is the series's sum.

Very nice and quite emphatic.

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• 3 weeks later...

cool

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