Q. The sum of the series $(1 + 2) + \left( 1 + 2 + 2^{2} \right) + \left( 1 + 2 + 2^{2} + 2^{3} \right) + ...upto\ n$ terms is
A) $2^{n + 2} - n - 4$
B) $2\left( 2^{n} - 1 \right) - n$
C) $2^{n + 1} - n$
D) $2^{n + 1} - 1$
Explanation
(a)
Here, $T_{n} = 1 + 2 + 2^{2} + 2^{3} + ... + 2^{n}$
$= \frac{1(2^{n + 1} - 1)}{2 - 1} = 2^{n + 1} - 1$
$\therefore\Sigma T_{n} = \Sigma\left( 2^{n + 1} - 1 \right) = \Sigma 2^{n + 1} - \Sigma 1$
$= \left( 2^{2} + 2^{3} + ... + 2^{n + 1} \right) - n$
$= 2\left( 2 + 2^{2} + ... + 2^{n} \right) - n$
$= \frac{4(2^{n} - 1)}{2 - 1} - n = 2^{n + 2} - n - 4$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.