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

Theorem

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

Direct Proof
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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:

$$1+2+\dots+k+(k-1)+\dots+1=k^2$$

Induction Step: Show true for $$n=k+1$$:

We need to show that $$1+2+\dots+(k+1)+k+(k-1)+\dots+1=(k+1)^2$$.

So:

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The result follows by induction.