# Different ways to write consecutive integer sum

$\begin{array} {l l l } 1 + 2 &=& 3 \\ 1 + 2 + 3 + \cdots + 14 &=& 15 + 16 + \cdots + 20 \\ 1 + 2 + 3 + \cdots + 84 &=& 85 + 86 + \cdots + 119 \end{array}$

The above 3 equations show that the sum of the first few consecutive positive integers can also be expressed as the sum of the subsequent (but fewer) consecutive integers.

What is the smallest integer $$n>84$$ such that $$1 + 2 + 3+ \cdots + n$$ can be expressed as $$(n+1) + (n+2) + \cdots + (n+m)$$ for some positive integer $$m$$?

Bonus: Generalize this.

×