Closed Form for Triangular Numbers/Proof using Odd Number Theorem
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Theorem
The closed-form expression for the $n$th triangular number is:
- $\ds T_n = \sum_{i \mathop = 1}^n i = \frac {n \paren {n + 1} } 2$
Proof
\(\ds \sum_{j \mathop = 1}^n \paren {2 j - 1}\) | \(=\) | \(\ds n^2\) | Odd Number Theorem | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \sum_{j \mathop = 1}^n \paren {2 j - 1} + \sum_{j \mathop = 1}^n 1\) | \(=\) | \(\ds n^2 + n\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \sum_{j \mathop = 1}^n \paren {2 j}\) | \(=\) | \(\ds n \paren {n + 1}\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \sum_{j \mathop = 1}^n j\) | \(=\) | \(\ds \frac {n \paren {n + 1} } 2\) |
$\blacksquare$
Sources
- 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {A}.2$: Pythagoras (ca. $\text {580}$ – $\text {500}$ B.C.)