Problem: Find a closed form for the sum
Solution: Consider the expansion of
It’s not hard to see that if we let the desired sum be , we have
since each pair is counted twice and the squared terms are there. But we can simplify the sums by well-known formulas to get
after massive simplification. QED.
Comment: Symmetry was definitely the best way to approach this problem (or as far as I know anyway). You could’ve found a lot of ugly summations to get to the desired expression as well.
Practice Problem: Can you generalize? In any way, shape, or form.
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