For a positive integer


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__________.


According to the question,

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

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

$\Rightarrow 2 n-7 r+2=0$ ...........(1)

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

$\Rightarrow 5 n-17 r-12=0$...........(2)

Solving eqns. (1) and (2),

$n=118, r=34$

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