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Let $$\{a_n\}$$ be a sequence of integers such that $$a_1 = 1$$, $$a_{m+n} = a_m + a_n + mn$$ for all positive integers $$m$$ and $$n$$. Then $$a_{12}$$ is
Putting $$m = 1$$ gives $$a_{n+1} = a_n + n + 1$$, so $$a_n = 1 + 2 + \cdots + n = \frac{n(n + 1)}{2}$$. Hence $$a_{12} = \frac{12 \times 13}{2} = 78$$.
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