# Closed Form for Triangular Numbers/Historical Note

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## Historical Note on Closed Form for Triangular Numbers

Some sources report that the closed-form expression for the $n$th triangular number was proved by Archimedes during the course of his proofs of the volumes of various solids of revolution in his *On Conoids and Spheroids*.

Others suggest that it was known as far back as Pythagoras of Samos, circa $550$ B.C.E.

## Sources

- 1980: David M. Burton:
*Elementary Number Theory*(revised ed.) ... (previous) ... (next): Chapter $1$: Some Preliminary Considerations: $1.3$ Early Number Theory: Problems $1.3$: $1 \ \text {(a)}$ - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {A}.5$: Archimedes (ca. $\text {287}$ – $\text {212}$ B.C.)