Closed Form for Triangular Numbers/Historical Note

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.)