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I need a little help in summing the the following series:

$$ 1+2v^4+3v^8+4v^{12}+\ldots + 20v^{76}?$$


Is there a closed formula for summing $$ \sum_{k=0}^{n} k\cdot ar^k?$$

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    $$\sum_{k=0}^{n-1}k\cdot ar^k=ar\sum_{k=1}^{n-1}kr^{k-1}=ar\frac{d}{dr}\left(\sum_{k=1}^{n-1}r^k\right)= \frac{ar(r-r^n)}{1-r}$$ It’s been derived many times here, but I don’t immediately find an example2012-11-05
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    There are various approaches, but you could try here setting $x=v^4$. Then sum $x+x^2+x^3+ \dots$ and then differentiate both sides.2012-11-05

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