For a positive integer n, (1+1x)^n is expanded in increasing powers of x.

Question

For a positive integer $n$, $\left(1+\dfrac1x\right)^n$ 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 \_\_\_\_\_\_\_\_.

Answer: 118

Accepted Answer

${}^nC_{r-1}:{}^nC_r:{}^nC_{r+1}=2:5:12$

$$\frac{{}^nC_r}{{}^nC_{r-1}}=\frac52\quad\Rightarrow\quad\frac{n-r+1}{r}=\frac52\quad\Rightarrow\quad2n-7r+2=0\quad\text{...(i)}$$

$$\frac{{}^nC_{r+1}}{{}^nC_r}=\frac{12}{5}\quad\Rightarrow\quad\frac{n-r}{r+1}=\frac{12}5\quad\Rightarrow\quad5n-17r-12=0\quad\text{...(ii)}$$

Solving (i),(ii): $n=118,\ r=34$

Solutions · 1