Closed Form for Triangular Numbers/Direct Proof/Historical Note

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Historical Note on Closed Form for Triangular Numbers: Direct Proof

This direct proof is the method supposedly employed by Carl Friedrich Gauss who, when very young (according to the apocryphal story), calculated the sum of the numbers from $1$ to $100$ before the teacher had barely sat back down after setting the assignment.

Whether this story is actually true or not is the subject of speculation.