Q. The interval in which $x$ must lie so that the greatest term in the expansion of ${(1 + x)}^{2n}$ has the greatest coefficient, is
A) $\left( \frac{n - 1}{n},\frac{n}{n - 1} \right)$
B) $\left( \frac{n}{n + 1},\frac{n + 1}{n} \right)$
C) $\left( \frac{n}{n + 2},\frac{n + 2}{n} \right)$
D) None of these
Explanation
(b)
Here, the greatest coefficient is $\ ^{2n}C_{n}$
$\therefore\ \ ^{2n}C_{n}x^{n} > \ ^{2n}C_{n + 1}x^{n - 1}\ \Rightarrow \ x > \frac{n}{n + 1}$
and $\ ^{2n}C_{n}x^{n} > \ ^{2n}C_{n - 1}x^{n - 1}\ \Rightarrow \ x < \frac{n + 1}{n}$
$\therefore x$ must lie in the interval $\left( \frac{n}{n + 1},\frac{n + 1}{n} \right)$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.