Given $p$,$x$ and $n$ , I have to calculate(find a closed form for the following four series) :
1) $p^1+p^2+p^3+.......+p^n$
2) $p^1(p^1-1) +p^2(p^2-1)+p^3(p^3-1)+.......+p^n(p^n-1)$
3) $p^1(p^1-1)(p^1-2)(p^1-3) +p^2(p^2-1)(p^2-2)(p^2-3)+p^3(p^3-1)(p^3-2)(p^3-3)+.......+p^n(p^n-1)(p^n-2)(p^n-3)$
4) $p^1(p^1-1)(p^1-2)(p^1-3)....(p^1-x) +p^2(p^2-1)(p^2-2)(p^2-3)........(p^2-x)...+p^3(p^3-1)(p^3-2)(p^3-3)......(p^3-x)+............+p^n(p^n-1)(p^n-2)(p^n-3)........(p^n-x)$
I was able to calculate the first two series using sum of G.P. method. Looking for the solution of last 2 series.
For the third one, just multiply out.
$$\begin{align*}\sum_{k=1}^n p^k(p^k-1)(p^k-2)(p^k-3) & = \sum_{k=1}^n\left(p^{4k}-6p^{3k}+11p^{2k}-6p^k\right) \\ & = \sum_{k=1}^n(p^4)^k-6\sum_{k=1}^n(p^3)^k+11\sum_{k=1}^n(p^2)^k-6\sum_{k=1}^np^k\end{align*}$$
Now, each is a geometric sum.