Generating Function of 3(5^n)-2n

Question

I am unsure as to whether or not I solved this generating function correctly. $a_n = 3(5^n) − 2n$ and the generating function is given by $$ \begin{split} &=3\sum 5^n - 2 \sum n\\ &=3(1+5+5^2+5^3...) -2(1+2+3+4…)\\ &= 3 \times \frac{5}{1-n} -2 \times \frac{n}{(1-n)^2}\\ &= \frac{15}{1-n} - \frac{2n}{(1-n)^2} \end{split} $$ Does that look correct?

Answer 1

Here is how I would do it. Let $a_n = 3 \cdot 5^n - 2n$ and then $$ \begin{split} A(x) &= \sum_{k=0}^\infty a_k x^k \\ &= 3 \sum_{k=0}^\infty 5^k x^k - 2 \sum_{k=0}^\infty k x^k\\ &= 3 \sum_{k=0}^\infty (5x)^k - 2x \sum_{k=1}^\infty k x^{k-1}\\ &= \frac{3}{1-5x} - 2x \frac{d}{dx} \left[\frac{1}{1-x}\right] \end{split} $$ Can you finish this?