Question

The ninth term in the expansion of ${\left\{3^{{\log }_3\sqrt{{25}^{x-1}+7}}+3^{-1/8{\log }_3\left(5^{x-1}+1\right)}\right\}}^{10}$ is equal to $180$, then $x$ is:

• A. $2$
• B. $1$
• C. $4$
• D. $Noneofthese$

Answer 1

The correct option is A $2$

Let $5^{x-1}=t$

$[\sqrt{t^2+7}+(t+1{)}^{\frac{-1}{8}}{]}^{10}$

8th term of the expansion is ${=}^{10}{C}_8(t^2+7)(t+1{)}^{-1}=45\times \frac{t^2+7}{t+1}=180$ (As given in the question)

$\Rightarrow t^2-4t+3=0\Rightarrow t=1,3$

If $t=1\Rightarrow x=1$