Q. The sum of $(x + 2)^{n - 1} + (x + 2)^{n - 2}(x + 1) + (x + 2)^{n - 3}(x + 1)^{2} + ... + (x + 1)^{n - 1}$ is equal to
A) $(x + 2)^{n - 2} - (x + 1)^{n}$
B) $(x + 2)^{n - 1} - (x + 1)^{n - 1}$
C) $(x + 2)^{n} - (x + 1)^{n}$
D) None of these
Explanation
(c)
Given, sum
$= (x + 2)^{n - 1}\left\{ 1 + \left( \frac{x + 1}{x + 2} \right) + \left( \frac{x + 1}{x + 2} \right)^{2} + ... + \left( \frac{x + 1}{x + 2} \right)^{n - 1} \right\}$
$= (x + 2)^{n - 1}\left\{ \frac{1 - \left( \frac{x + 1}{x + 2} \right)^{n}}{1 - \left( \frac{x + 1}{x + 2} \right)} \right\}$
$= \frac{(x + 2)^{n - 1}\left\{ (x + 2)^{n} - (x + 1)^{n} \right\} \bullet (x + 2)}{(x + 2)^{n}}$
$= (x + 2)^{n} - (x + 1)^{n}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.