Question

If the sum to infinity of the series $1+2r+3r^2+4r^3+\dots$ is $9/4$, then value of $r$ is

• A. 1/2
• B. 1/3
• C. 1/4
• D. None of these

Answer 1

Answer: (B)

$S=1+2r+3r^2+4r^3+\dots$
$rS=r+2r^2+3r^3+4r^4+\dots$
Subtracting,
$(1-r)S=1+r+r^2+r^3+\dots$
$=\frac{1}{1-r}$
$\Rightarrow S=\frac{1}{(1-r{)}^2}$
Given $S=9/4\Rightarrow \frac{1}{(1-r{)}^2}=9/4$
$\Rightarrow 1-r=\pm \frac{2}{3}$
$\Rightarrow r=1/3$ or $5/3$
Hence $r=1/3$ as $-1<r<1$