Q. In the expansion of the following expression $1 + (1 + x) + (1 + x)^{2} + ... + (1 + x)^{n}$, the coefficient of $x^{4}(0 \leq k \leq n)$ is
A) $\ ^{n + 1}C_{k + 1}$
B) $\ ^{n}C_{k}$
C) $\ ^{n}C_{n - k - 1}$
D) None of these
Explanation
(a)
The given expression is
$1 + (1 + x) + (1 + x)^{2} + ... + (1 + x)^{n}$ being in GP
Let $S = 1 + (1 + x) + (1 + x)^{2} + ... + (1 + x)^{n}$
$= \frac{(1 + x)^{n + 1} - 1}{(1 + x) - 1} = x^{- 1}\left\lbrack (1 + x)^{n + 1} - 1 \right\rbrack$
$\therefore$ The coefficient of $x^{k}$ in $S$
$=$ The coefficient of $x^{k}$ in $\left\lbrack (1 + x)^{n + 1} - 1 \right\rbrack$
$= \ ^{n + 1}C_{k + 1}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.