For a positive integer


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

in increasing powers of $x$. If three consecutive coefficients in this expansion are in the ratio, $2: 5: 12$, then $\mathrm{n}$ is equal to


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

Now $\frac{{ }^{n} \mathrm{C}_{\mathrm{r}-1}}{{ }^{n} \mathrm{C}_{r}}=\frac{2}{5}$

$\Rightarrow 7 \mathrm{r}=2 \mathrm{n}+2$ .......(1)

$\frac{{ }^{n} \mathrm{C}_{r}}{{ }^{n} \mathrm{C}_{r+1}}=\frac{5}{12}$

$\Rightarrow 17 \mathrm{r}=5 \mathrm{n}-12$......(2)

On solving (1) & (2)

$\Rightarrow n=118$

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