A possible value of


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 :


  1. 0

  2. $-1$

  3. 2

  4. 1

Correct Option: , 4


${ }^{10} \mathrm{C}_{8}\left(25^{(\mathrm{x}-1)}+7\right) \times\left(5^{(\mathrm{x}-1)}+1\right)^{-1}=180$

$\Rightarrow \frac{25^{x-1}+7}{5^{(x-1)}+1}=4$

$\Rightarrow \frac{t^{2}+7}{t+1}=4$

$\Rightarrow t=1,3=5^{x-1}$

$\Rightarrow x-1=0$ (one of the possible value)

$\Rightarrow x=1$


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