Question

The co-efficient of $x^{401}$ in the expansion of ${(1+x+x^2+\cdot \cdot \cdot +x^9)}^{-1}$ $(\left|x\right|<1)$ is

• A. $1$
• B. $-1$
• C. $2$
• D. $-2$

Answer 1

Answer: (B)

$-1$

Given expression is $1+x+x^2+...x^9$
$=\frac{1-x^{10}}{1-x}$
Now
$(\frac{1-x^{10}}{1-x}{)}^{-1}$
$=(1-x)(1-x^{10}{)}^{-1}$
$=(1-x)(1+x^{10}+x^{20}+...x^{400}+..\infty )$
Hence coefficient of $x^{401}$
$=-1$