Question

The coefficient of $ {{x}^{n}} $ in the expansion of $ {{(1-9x+20{{x}^{2}})}^{-1}} $

• A. $ {{5}^{n}}-{{4}^{n}} $
• B. $ {{5}^{n+1}}-{{4}^{n+1}} $
• C. $ {{5}^{n-1}}-{{4}^{n-1}} $
• D. None of these

Answer 1

Answer: (B)

We have, $\left(1-9 x+20 x^{2}\right)^{-1}=[(1-5 x)(1-4 x)]^{-1}$ $=\frac{1}{(1-5 x)(1-4 x)}=\frac{5}{1-5 x}-\frac{4}{1-4 x}$
$=5(1-5 x)^{-1}-4(1-4 x)^{-1}$
$=5\left[1+5 x+(5 x)^{2}+\ldots+(5 x)^{n}+\ldots\right]$
$-4\left[1+4 x+(4 x)^{2}+\ldots+(4 x)^{n}+\ldots\right] \Rightarrow $ The coefficient of $x^{n}$ is $5^{n+1}-4^{n+1}$