So I got this question
A sequence is given by $a_0=1, \ a_1= 11$ and $a_n = 8a_{n-1} - 15a_{n-2}$. We want a closed expression of $a_n$, and express the formal power series $A(x)= \sum_{n=0}^{\infty} a_nx^n$ as a quotient of two polynomials.
Calculating $a_n$ is easy, by solving the second degree equation and solving for the constans we get that $a_n = -3^{n+1} +4*5^n$. But how exactly can I get a power series expansion of two polynomials from this?
Thanks in advance!
It is required to use nothing but the geometric series: $$\sum_{n=0}^{+ \infty} (-3^{n+1}+4 \cdot 5^n) x^n = -3\sum_{n=0}^{+ \infty} (3x)^n + 4 \sum_{n=0}^{+ \infty} (5x)^n = \frac{-3}{1-3x}+\frac{4}{1-5x}$$
We have $A(x) - a_0 - a_1 x = 8(xA(x)-a_1 x) - 15 x^2 A(x)$ and so $$ A(x) = \frac{3x+1}{(3x-1)(5x-1)} $$ There is no need to find $a_n$ in closed form first.
Expanding the quotient in partial fractions will give you two geometric series, from which you can find $a_n$ in closed form.