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.
 A: 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}
A: $$a_n=2^n+3^n,\\
a_{n+1}=2\cdot2^n+3\cdot3^n,\\
a_{n+2}=4\cdot2^n+9\cdot3^n$$
Then we eliminate $2^n$ and $3^n$ from the above relations by
$$a_{n+2}-4a_n=5\cdot3^n,\\
a_{n+1}-2a_n=3^n$$
and
$$a_{n+2}-4a_n=5(a_{n+1}-2a_n)$$
or
$$a_{n+2}=5a_{n+1}-6a_n.$$

You can reach the same conclusion by noting that the characteristic polynomial has roots $2$ and $3$, hence is $$r^2-5r+6,$$ corresponding to the recurrence
$$a_{n+2}-5a_{n+1}+6a_n=0.$$
A: Hint: What are the roots of the characteristic equation of that recursive relation?
A: Looking very closely at the terms one can identify
$$a_0=2$$
$$a_1=5=3*a_0-1=3a_0-2^0$$
$$a_2=13=3a_1-2=3a_1-2^1$$
$$a_3=35=3a_2-4=3a_2-2^2$$
$$a_4=97=3a_3-8=3a_3-2^3$$
$$a_5=275=3a_4-16=3a_4-2^4$$
And in general:
$$a_{n+1}=3a_n-2^n$$
