What math statement with proof do you find most beautiful and elegant, where such is accessible to a general audience, meaning you could state, prove, and explain it to a general audience in roughly 5 \pm\epsilon minutes.

Let’s define ‘general audience’ as approximately an average adult with education and experience comparable to someone holding a bachelor’s degree in any non science major (e.g. history) from an average North American university.

**Answer**

I really like the proof of \sum_{i=1}^n i = \dfrac{n(n+1)}{2} in which 1 + 2 + \cdots + (n-1) + n is written forwards then backwards and summed. It is claimed that Gauss had come up with this when he was just a child, although contested.

**The proof**

Let s = 1 + 2 + \cdots + (n-1) + n.

Clearly,

s = n + (n-1) + \cdots + 2 + 1.

Sum to get 2s = \underbrace{(n+1) + (n+1) + \cdots + (n+1) + (n+1)}_{n \text{ times}}.

Hence, 2s = n(n+1), and s = \dfrac{n(n+1)}{2}.

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