A) \[\frac{1-{{x}^{n}}}{1-x}\]
B) \[\frac{x(1-{{x}^{n}})}{1-x}\]
C) \[\frac{n(1-x)-x(1-{{x}^{n}})}{{{(1-x)}^{2}}}\]
D) None of these
Correct Answer: C
Solution :
\[1+(1+x)+(1+x+{{x}^{2}})+...\]+ \[(1+x+{{x}^{2}}+{{x}^{3}}+...+{{x}^{n-1}})+...\] Required sum =\[\frac{1}{(1-x)}\left\{ \,(1-x)+(1-{{x}^{2}})+(1-{{x}^{3}}) \right.\] \[\left. +(1-{{x}^{4}})+..........\text{uptp}\ n\ \text{terms} \right\}\] \[=\frac{1}{(1-x)}[n-\{x+{{x}^{2}}+{{x}^{3}}+..........\text{upto}\ n\ \text{terms }\!\!\}\!\!\text{ }\,]\] \[=\frac{1}{(1-x)}\left[ n-\frac{x(1-{{x}^{n}})}{1-x} \right]=\frac{n(1-x)-x(1-{{x}^{n}})}{{{(1-x)}^{2}}}\].
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