Citation/Reference for this Formula

Question

Does anyone know any sources that could possibly be used as a citation of some form of the formula below?

$$\sum_{n=0}^{k} (a+n)(b-n) = (k+1)ab + \frac{k(k+1)}{2}(b-a)+\frac{k(k+1)(2k+1)}{6}$$

Answer 1

Once you've fixed a couple of minor typos that make it false in its current form, it's an essentially-trivial consequence of some very well known expansions, so I'd just cite your favourite sources for those, then prove it. In particular, $(a+n)(b-n) = ab + (b-a)n - n^2$, so

\begin{align*}\sum\limits_{n=0}^k(a+n)(b-n) &= \left(ab\sum\limits_{n=0}^k1\right) + (b-a)\sum\limits_{n=0}^kn - \sum\limits_{n=0}^kn^2 \\&=ab(k+1) + (b-a)\frac{k(k+1)}{2} - \frac{k(k+1)(2k+1)}{6}.\end{align*}

(Notice the $b$ replacing your second $a$, and the $-$ instead of $+$ before the final term)

Where the summations are the obvious sum of $k+1$ constants, the standard sum of the first $k$ positive integers, and the standard sum of the first $k$ squares of positive integers.