Question

Let \((a_k)\) be a sequence of integers such that \(a_1 = 1\) and \(a_{m + n} = a_m + a_n + mn\) for all positive integers \(m\) and \(n\). Find \(a_{12}\).

Answer 1

Let \(m=1\),
\[a_{n+1} = a_n+n+1 \\\implies a_{n+1}-a_n = n+1\\\implies \sum\limits_{n=1}^{11} (a_{n+1}-a_n) = \sum\limits_{n=1}^{11} (n+1)\\
\]

left side telescopes and right side can be easily evaluated