1+2+...+n+(n-1)+...+1 = n^2/Proof 3

Proof
Proof by induction:

Basis for the Induction
$n = 1$ holds trivially.

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


 * $1 + 2 + 1 = 4$

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

So shown for basis for the induction.

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

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

Induction Step
This is our induction step:

The result follows by induction.