Q. The AM of the series $1,\ 2,\ 4,\ 8,\ 16,\ ...,2^{n}$ is
A) $\frac{2^{n} - 1}{n}$
B) $\frac{2^{n + 1} - 1}{n + 1}$
C) $\frac{2^{n} + 1}{n}$
D) $\frac{2^{n} - 1}{n + 1}$
Explanation
(b)
The required AM is
$\overline{X} = \frac{1 + 2 + 2^{2} + 2^{3} + ... + 2^{n}}{n + 1}$
$= \frac{1\left( 2^{n + 1} - 1 \right)}{(2 - 1)}.\frac{1}{(n + 1)} = \frac{2^{n + 1} - 1}{n + 1}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.