Question:
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
Solution:
${ }^{n} C_{\Gamma-1}:{ }^{n} C_{r}:{ }^{n} C_{n+1}=2: 5: 12$
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)
$\Rightarrow n=118$