Finding coefficient of $x^n$ in $(1-9x+20x^2)^{-1}$.

Question

Well I tried to factorize it and simplify it to $(1-5x)^{-1}$ multiply by $(1-4x)^{-1}$. Next I try to expand it and see what terms to multiply to get $x^n$ coefficient but it's getting really messy. Kindly show how to proceed further or better suggest an altogether different approach.

Answer 1

$$(1-5x)^{-1}(1-4x)^{-1}=\left(\sum_{r=0}^\infty(5x)^r\right)\left(\sum_{t=0}^\infty(4x)^t\right)$$ for $$|4x|,|5x|<1$$

So, the coefficient of $x^n$ will be

$$\sum_{r=0}^n 5^{n-r}4^r=5^n\cdot\dfrac{1-\left(\dfrac45\right)^{n+1}}{1-\dfrac45}=?$$

Answer 2

hint...express $\frac{1}{(1-4x)(1-5x)}$ in partial fractions then use the Binomial Theorem or geometric series