Q. If $|a| < 1$, then $1 + 2a + 3a^{2} + 4a^{3} + ...$ is equal to
A) $\frac{1}{1 - a}$
B) $\frac{1}{1 + a}$
C) $\frac{1}{1 + a^{2}}$
D) $\frac{1}{(1 - a)^{2}}$
Explanation
(d)
Given series, is a arithmetic geometric series.
Here, $a_{1} = 1,\ \ d = 1,\ \ \ r = a$
$\therefore S_{\infty} = \frac{a_{1}}{1 - r} + \frac{d.r}{(1 - r)^{2}}$
$= \frac{1}{1 - a} + \frac{1.a}{(1 - a)^{2}} = \frac{1}{(1 - a)^{2}}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.