For a positive integer $n$ the binomial expression ${(1 + \frac{1}{x})}^{n}$ 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

Open in App

Solution

Let the $3$ consecutive coefficients are $n_{C_{r - 1}} : n_{C_{r}} : n_{C_{r + 1}} :: 2 : 5 : 12$

Here $\frac{n_{C_{r - 1}}}{n_{C_{r}}} = \frac{2}{5}; \frac{n_{C_{r}}}{n_{C_{r + 1}}} = \frac{5}{12}$
$\frac{r}{n - r + 1} = \frac{2}{5}; \frac{r + 1}{(n - r)} = \frac{5}{12}$
$7 r = 2 n + 2; 17 r = 5 n - 12$
$\Rightarrow \frac{2 n + 2}{7} = \frac{5 n - 12}{17}$
$\Rightarrow 34 n + 34 = 35 n - 84$
$\Rightarrow n = 118$

Suggest Corrections

Similar questions

n

(1 + x)^{n + 5}

5 : 10 : 14,

If n is positive integer and three consecutive coefficients in the expansion of $(1 + x)^{n}$ are in the ratio 6 : 33 : 110, then n =

If n is positive integer and three consecutive

coefficients in the expansion of $(1 + x)^{n}$ are in the

ratio 6 : 33 : 110, then n =

(x + 1)^{n}

x

2 : 15 : 70

(x + 1)^{n}

x

2 : 15 : 70

Join BYJU'S Learning Program