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:

$\displaystyle T_n = \sum_{i \mathop = 1}^n i = \frac {n \paren {n + 1} } 2$


Proof

\(\displaystyle \sum_{j \mathop = 1}^n \left({2j - 1}\right)\) \(=\) \(\displaystyle n^2\) Odd Number Theorem
\(\displaystyle \leadsto \ \ \) \(\displaystyle \sum_{j \mathop = 1}^n \left({2j - 1}\right) + \sum_{j \mathop = 1}^n 1\) \(=\) \(\displaystyle n^2 + n\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle \sum_{j \mathop = 1}^n \left({2 j}\right)\) \(=\) \(\displaystyle n \left({n + 1}\right)\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle \sum_{j \mathop = 1}^n j\) \(=\) \(\displaystyle \frac {n \left({n + 1}\right)} 2\)

$\blacksquare$


Sources