For a positive integer n, (1+1 x )^ n is expanded in increasing p… - Vidyadip

MCQ

For a positive integer $n,\left(1+\frac{1}{x}\right)^{n}$ 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

A
$115$
B
$128$
C
$138$
✓
$118$

✓

Answer

Correct option: D.

$118$

d
${ }^{ n } C _{ r -1}:{ }^{ n } C _{ r }:{ }^{ n } C _{ r +1}=2: 5: 12$

Now $\frac{{ }^{n} C_{r-1}}{{ }^{n} C_{r}}=\frac{2}{5}$

$\Rightarrow 7 r=2 n+2$

$\frac{{ }^{n} C_{r}}{{ }^{n} C_{r+1}}=\frac{5}{12}$

$\Rightarrow 17 r =5 n -12$

On solving (1)$\&(2)$

$\Rightarrow n =118$

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