Question

If the sum of the series $1+2r+3r^2+4r^3+\cdots \infty$ is $\frac{9}{4},$ then the value of $r$ is

• A. $\frac{1}{2}$
• B. $\frac{1}{3}$
• C. $\frac{1}{4}$
• D. $\frac{5}{3}$

Answer 1

The correct option is B $\frac{1}{3}$

$S=1+2r+3r^2+4r^3+\dots$
$rS=r+2r^2+3r^3+4r^4+\dots$
$\Rightarrow (1-r)S=1+r+r^2+r^3+\dots$
$\Rightarrow (1-r)S=\frac{1}{1-r}$ provided $|r|<1$
$\Rightarrow S=\frac{1}{(1-r{)}^2}$

Given, $S=\frac{9}{4}$
$\Rightarrow \frac{1}{(1-r{)}^2}=\frac{9}{4}\Rightarrow 1-r=\pm \frac{2}{3}\Rightarrow r=\frac{1}{3},\frac{5}{3}\therefore r=\frac{1}{3}(\because |r|<1)$