Question

A possible value of $x$, for which the ninth term in the expansion of ${\{3^{{\log }_3\sqrt{{25}^{x-1}+7}}+3^{(-\frac{1}{8}){\log }_3(5^{x-1}+1)}\}}^{10}$ in the increasing powers of $3^{(-\frac{1}{8}){\log }_3(5^{x-1}+1)}$ is equal to $180$, is

• A. $-1$
• B. $0$
• C. $1$
• D. $2$

Answer 1

The correct option is C $1$

Given expression reduces to
${\left[\right(5^{2(x-1)}+7{)}^{\frac{1}{2}}+(5^{x-1}+1{)}^{-\frac{1}{8}}]}^{10}$
Ninth term is
${}^{10}{C}_8(5^{2(x-1)}+7)(5^{x-1}+1{)}^{-1}=180$
Let $5^{x-1}=t$
$(t^2+7)(t+1{)}^{-1}=4$
$\Rightarrow t^2+7=4t+4\Rightarrow t^2-4t+3=0\Rightarrow (t-3)(t-1)=0$
$\Rightarrow 5^{x-1}=1$ or $3$
$x=1\text{or}x=1+{\log }_{5}3$