Question:
A possible value of ' $x$ ', for which the ninth term in
the expansion of $\left\{3^{\log _{3} \sqrt{25^{x-1}+7}}+3^{\left(-\frac{1}{8}\right) \log _{3}\left(5^{x-1}+1\right)}\right\}^{10}$ in
the increasing powers of $3^{\left(-\frac{1}{8}\right) \log _{3}\left(5^{x-1}+1\right)}$ is equal to 180 , is :
Correct Option: , 4
Solution:
${ }^{10} \mathrm{C}_{8}\left(25^{(\mathrm{x}-1)}+7\right) \times\left(5^{(\mathrm{x}-1)}+1\right)^{-1}=180$
$\Rightarrow \frac{25^{x-1}+7}{5^{(x-1)}+1}=4$
$\Rightarrow \frac{t^{2}+7}{t+1}=4$
$\Rightarrow t=1,3=5^{x-1}$
$\Rightarrow x-1=0$ (one of the possible value)
$\Rightarrow x=1$
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