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The coefficient if $x^n$ in the expansion of $(1+x+x^2+...)^{-n}$ is

$\begin{array}{1 1}(A)\;1\\(B)\;(-1)^n\\(C)\;n\\(D)\;n+1\end{array} $

1 Answer

Toolbox:
  • $T_{r+1}=nC_ra^{n-r} b^r$
$(1+x+x^2+.......)^{-n}=(\large\frac{1}{1-x})^n$
$\Rightarrow (1-x)^n$
$\therefore$ Term containing $x^n=nC_n (-x)^n$
$\Rightarrow (-1)^n x^n$
$\therefore$ Coefficient of $x^n=(-1)^n$
Hence (B) is the correct answer.
answered Jun 25, 2014 by sreemathi.v
 

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