Q. The sum of $n$ terms of the series $1 + (1 + x) + \left( 1 + x + x^{2} \right) + \ ...$ will be
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
Explanation
(c)
Required sum
$= \frac{1}{1 - x}\lbrack(1 - x) + \left( 1 - x^{2} \right) + \left( 1 - x^{3} \right) + ...\$up to $n$ terms]
$= \frac{1}{1 - x}\left\lbrack n - (x + x^{2} + x^{3} + \ldots upto\ n\ terms) \right\rbrack$
$= \frac{1}{1 - x}\left\lbrack n - x\frac{\left( 1 - x^{n} \right)}{1 - x} \right\rbrack = \frac{n(1 - x) - x(1 - x^{n})}{(1 - x)^{2}}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.