Finding a formula to a given $\sum$ using generating functions Find a close formula to the sum $\sum_{k=0}^{n}k\cdot 5^k$
I tried using generating functions using the differences sequences with no luck.

Set $g_k=\frac{1}{4}5^{k+1}$.  Note that $\Delta g_k=g_{k+1}-g_k=\frac{1}{4}(5^{k+1}-5^k)=5^k$.  Set $f_k=k$; we have $\Delta f_k=f_{k+1}-f_k=1$.  The partial summation formula gives:
We have $$\sum_0^nk5^k=\sum_0^nf_k\Delta g_k=(f_{n+1}g_{n+1}-f_0g_0)-\sum_0^ng_{k+1}\Delta f_k$$
The RHS simplifies to $$(n+1)\frac{1}{4}5^{n+2}-0-\sum_{0}^n \frac{1}{4}5^{k+2}=\frac{1}{4}\left((n+1)5^{n+2}-5^2\frac{5^{n+1}-1}{5-1}\right)=\frac{25}{4}\left((n+1)5^n-\frac{1}{4}(5^{n+1}-1)\right)=\frac{25}{4}\left(\left(n-\frac{5}{4}\right)5^n+\frac{1}{4}\right)$$
