Mathematics - Binomial Theorem Question with Solution | TestHub

Let coefficient of first term is ${}^{n}C_r$ 

$\frac{{\displaystyle {}^{n}C_r}}{{\displaystyle {}^{n}C_{r+1}}}=\frac{1}{7}$

$\Rightarrow \frac{r+1}{n-r}=\frac{1}{7}$

$\Rightarrow 7r+7=n-r$

$n-8r=7 \dots \left(1\right)$ 

Also,  $\frac{{\displaystyle {}^{n}C_{r+1}}}{{\displaystyle {}^{n}C_{r+2}}}=\frac{7}{42}=\frac{1}{6}$

$\Rightarrow \frac{r+2}{n-r-1}=\frac{1}{6}$

$\Rightarrow 6r+12=n-r-1$

$n-7r=13 \dots \left(2\right)$ 

Solving $\left(1\right)$ and $\left(2\right)$, we get $r=6$

Hence, $7^{\mathrm{th}}$ term is the answer.