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