Suppose that $a_n = a_0r^n$. This is a geometric sequence, since $a_{n+1}/a_n = r$ is independent of n.
a) Give the generating function $A(X) = \sum_{n}a_nX^n$
b) Give the generating function $B(X) = \sum_{n}b_nX^n$ with $b_n = \sum_{k=0}^n a_k$
c) Calculate $[X^n]B(X)$, and thus give a formula for the sum of a finite geometric sequence.
For Part a) what I thought to do from my notes was
$A(X) + A(−X) = (a_0 + a_1X + a_2X^2 + a_3X^3 +...) +(a_0 + a_1(−X) + a_2(−X)^2 + a_3(−X)^3 +...) $
$= (a_0 + a_1X + a_2X^2 + a_3X^3 +...) + (a_0 − a_1X + a_2X^2 − a_3X^3 +...)$
$= (2a_0 + a_2X^2 + a_4X^4 +...) $
Which gives the generating function for the even terms and you can use it to get the odd terms with a small change. Im not sure if this is right however.