How to convert this sum into an equation?
$$ \sum_{i=1}^{C-1} i(R-i)(C-i) $$
Expanded:
$$ \sum_{i=1}^{C-1} (RCi - Ri^2 -Ci^2 + i^3) $$
Which can be represented as:
$$ RC\sum_{i=1}^{c-1}i - R\sum_{i=1}^{c-1}i^2-C\sum_{i=1}^{c-1}i^2 + \sum_{i=1}^{c-1}i^3 $$
Using the Faulhaber formula and having n = c-1, it can be represented as:
$$ {1\over 2}RC(n^2+n) -{1\over6}R(2n^3+3n^2+n) - {1\over 6}C(2n^3+3n^2+n)+{1\over4}(n^4+2n^3+n^2) $$
But the results from the sum and the formula from the Faulhaber formula are different. Where is my error?
Expand the product, which will give terms in $i, i^2,$ and $i^3$. Do you know how to sum each of those by Faulhaber's formula?