Have no idea where to go. A friend suggested taking the difference of Gauss sums, not sure what that means:
$$\sum_{i = 1}^{n}\sum_{j = i}^{2i}j$$
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HINT
Recall that $s(n) = \sum_{k=1}^n k = n(n+1)/2$ and note that $$\sum_{j=i}^{2i}j = \sum_{j=1}^{2i} j - \sum_{j=1}^{i-1} j = s(2i) - s(i-1) $$
gt6989b
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Why can you break it apart like that? May I have a further explanation? – user7252850 Sep 08 '17 at 00:57
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@user7252850 you add and subtract the terms from 1 to $i-1$ (i.e. $s(i-1)$) and the result is all terms from 1 to $2i$... – gt6989b Sep 08 '17 at 01:02