Find the coefficient of `x^n` in the expansion of `(1-9x+20 x^2)^(-1)dot`

Options

A. `5^(n) - 4^(n)`
B. `5^(n+1) - 4^(n+1)`
C. `5^(n -1) - 4^(n-1)`
D. none of these

Answer

b

Explanation

We have,
`(1 - 9x + 20 x^(2) ^(-1) = {(1 - 5x)(1 - 4 x )}^(-1) = (1)/((1 - 5x)(1 - 4x))`
`rArr (1 - 9x + 20 x^(2))^(-1) = (5)/(1-5x) - (4)/(1-4x)`
`rArr (1 - 9x + 20x^(2))^(-1) = 5 (1 - 5x)^(-1) - 4 (1- 4x)^(-1)`
`therefore ` Coefficient of `x^(n)` in `(1-9x + 20 x^(2))^(-1)`
`= 5 (5^(n)) - 4 (4^(n)) = 5^(n+1) - 4 ^(n+1)`.