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Split mixed sigma sums into arithmetic and geometric parts

Find (i) ∑k=0n(k+n+1)\sum_{k=0}^n(k+n+1) in terms of nn; (ii) ∑r=1N(4r+2−r)\sum_{r=1}^N(4r+2^{-r}) in terms of NN; (iii) ∑k=130(a2k+ka)\sum_{k=1}^{30}(a^{2k}+ka) in terms of aa.

CJC · Discussion · Question 9

Separate each summand into familiar sums and count its actual number of terms.

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