<s>[INST] Find generating function of series $a_{n} = 2^{n} + 3^{n} $ Find generating function of series $a_{n} = 2^{n} + 3^{n} $
I'm having a problem with this because at first i have to find recursive equation for $a_{n}$ . I find that $a_{0} = 2$ , $a_{1} = 5$ but how do I find the rest? Also I can't simply add generating functions, I have to solve recursive equation using generating functions. First step is to find recursive equation for $a_{n}$ then some other stuff that I can handle.
 [/INST] Hint:
$$\sum_{n=0}^\infty a_nx^n = \sum_{n=0}^\infty (2^n+5^n)x^n = \sum_{n=0}^\infty (2x)^n + \sum_{n=0}^\infty (5x)^n = \frac1{1-2x} + \frac1{1-5x}$$

Now you can proceed to find the recursive relation:
$$\sum_{n=0}^\infty a_nx^n = \frac1{1-2x} + \frac1{1-5x} = \frac{2-7x}{1-7x+10x^2}$$
$$\sum_{n=0}^\infty a_nx^n - 7\sum_{n=1}^\infty a_{n-1}x^n + 10 \sum_{n=2}^\infty a_{n-2}x^n = (1-7x+10x^2)\sum_{n=0}^\infty a_nx^n = 2-7x$$
Comparing powers gives:
\begin{align}
x^0:& \quad a_0 =2\\
x^1:& \quad a_1 - 7a_0 = -7 \implies a_1 = 7\\
x^k \text{ for } k \ge 2:& \quad a_k - 7a_{k-1} + 10a_{k-2} = 0
\end{align}
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