First slide

Geometric progression

Question

Sum of the series : $(1 + x) + (1 + x + x^{2}) +$ $(1 + x + x^{2} + x^{3}) + \dots$ upto n terms is

Easy

A

$\frac{1}{1 - x} [n - \frac{x^{2} (1 - x^{n})}{1 - x}]$
B

$\frac{1}{1 - x} [n - \frac{x^{3} (1 - x^{n})}{1 - x}]$
C

$\frac{1}{1 - x} [n - \frac{x (1 - x^{n})}{1 - x}]$
D

None of these

Solution

We have,
$(1 + x) + (1 + x + x^{2}) + (1 + x + x^{2} + x^{3})$ $+ \dots upto n terms$
$= \frac{1}{1 - x} [(1 + 1 + 1 + \dots$ n terms) $- (x^{2} + x^{3} + x^{4} + \dots to n terms)]$
$= \frac{1}{1 - x} [n - \frac{x^{2} (1 - x^{n})}{1 - x}]$

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