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

Theorem

 * $\forall n \in \N: 1 + 2 + \cdots + n + \left({n-1}\right) + \cdots + 1 = n^2$

Illustration

 * 1plus2plusnplus2plus1.png

Proof by Induction
Proof by induction:

Base case
$n = 1$ holds trivially.

Just to make sure, we try $n = 2$:


 * $1 + 2 + 1 = 4$

Likewise $n^2 = 2^2 = 4$.

So shown for base case.

Induction Hypothesis
This is our induction hypothesis:
 * $1 + 2 + \cdots + k + \left({k-1}\right) + \cdots + 1 = k^2$

Now we need to show true for $n=k+1$:
 * $1 + 2 + \cdots + \left({k + 1}\right) + k + \left({k-1}\right) + \cdots + 1 = \left({k + 1}\right)^2$

Induction Step
This is our induction step:

The result follows by induction.