1) What is the generating function of $a_n = n2^n, n\geq0$?
My answer: $f(x) = \sum a_nx^n = \sum n2^nx^n = \sum n(2x)^n$, but I have no idea where to go from here.
2) Let the sequence $s_n = a_0 + a_1 + a_2 + ... + a_n$. What is its generating function?
My answer: $f(x) = \sum a_nx^0 = \sum a_n ....$ this is where I am stuck.
Hint: We want $\sum n(2x)^n$. Note that $$1+2x+(2x)^2+ (2x)^3+\cdots=\frac{1}{1-2x}$$ for suitable $x$. Differentiate term by term, and you are almost there.