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7.Binomial Theorem
medium
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
$138$
D
$118$
(JEE MAIN-2020)
Solution
${ }^{ n } C _{ r -1}:{ }^{ n } C _{ r }:{ }^{ n } C _{ r +1}=2: 5: 12$
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)$\&(2)$
$\Rightarrow n =118$
Standard 11
Mathematics
