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

Direct Proof 1
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Direct Proof 2
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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:

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The result follows by induction.