For a positive integer $n,\left(1+\frac{1}{x}\right)^{n}$ is expanded in increasing powers of $x$. I

Question

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

Professor PC Thomas Classes · Answer

(118) : We have, $\left(1+\frac{1}{x}ight)^{n}$ Since three consecutive coefficient are in the ratio $2: 5: 12$.\ $	herefore \quad{ }^{n} C_{r-1}:{ }^{n} C_{r}:{ }^{n} C_{r+1}=2: 5: 12$\ $\Rightarrow \quad \frac{{ }^{n} C_{r}}{{ }^{n} C_{r-1}}=\frac{5}{2}$ and $\frac{{ }^{n} C_{r+1}}{{ }^{n} C_{r}}=\frac{12}{5}$\ $\Rightarrow \quad \frac{n-r+1}{r}=\frac{5}{2}$ and $\frac{n-r}{r+1}=\frac{12}{5}$\ $\Rightarrow \quad 2 n-2 r+2=5 r$ and $5 n-5 r=12 r+12$\ $\Rightarrow 2 n-7 r+2=0$ and $5 n-17 r-12=0$\ $\Rightarrow \quad \frac{n}{84+34}=\frac{r}{10+24}=\frac{1}{-34+35}$\ $\Rightarrow n=118$ and $r=34$\