A) \[\frac{1}{1-x}\]
B) \[\frac{1}{1+x}\]
C) \[\frac{1}{{{(1+x)}^{2}}}\]
D) \[\frac{1}{{{(1-x)}^{2}}}\]
Correct Answer: D
Solution :
This is an A.G.P. Let \[S=1+2x+3{{x}^{2}}+.......\infty \] \[\Rightarrow \]\[x.S=x+2{{x}^{2}}+........\infty \] Subtracting \[(1-x)S=1+x+{{x}^{2}}+.........\infty =\frac{1}{1-x}\] \[\therefore \]\[S=\frac{1}{{{(1-x)}^{2}}}\]. Aliter : Use \[S=\left[ 1+\frac{r}{1-r}\times \text{diff}\text{.}\ \text{of}\ \text{A}\text{.P}\text{.} \right]\frac{1}{1-r}\].
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