Q45 / 75 · JEE Main 2020 (02 Sep Shift 2) · Mathematics

For a positive integer $n, {(1 + \frac{1}{x})}^{n}$ is expanded in increasing powers of $x$ . If three consecutive coefficients in this expansion are in the ratio, $2 : 5 : 12,$ then $n$ is equal to

Correct answer: 118

Solution

${}^{n}C_{r - 1} : {}^{n}C_{r} : {}^{n}C_{r + 1} = 2 : 5 : 12$

$\Rightarrow \frac{{}^{n}C_{r}}{{}^{n}C_{r - 1}} = \frac{5}{2}$ and $\frac{{}^{n}C_{r + 1}}{{}^{n}C_{r}} = \frac{12}{5}$

$\Rightarrow \frac{n - r + 1}{r} = \frac{5}{2}$ and $\frac{n - r}{r + 1} = \frac{12}{5}$

$\Rightarrow 2 n - 7 r + 2 = 0$ and $5 n - 17 r - 12 = 0$

On solving $n = 118$ and $r = 34$

Hence, the correct answer is 118.