A possible value of 'x', for which the ninth term in the expansion of $${\left\{ {{3^{{{\log }_3}\sqrt {{{25}^{x - 1}} + 7} }} + {3^{\left( { - {1 \over 8}} \right){{\log }_3}({5^{x - 1}} + 1)}}} \right\}^{10}}$$ in the increasing powers of ${3^{\left( { - {1 \over 8}} \right){{\log }_3}({5^{x - 1}} + 1)}}$ is equal to 180, is :
Solution
${}^{10}{C_8}({25^{(x - 1)}} + 7) \times {({5^{(x - 1)}} + 1)^{ - 1}} = 180$<br><br>$\Rightarrow {{{{25}^{x - 1}} + 7} \over {{5^{(x - 1)}} + 1}} = 4$<br><br>$\Rightarrow {{{t^2} + 7} \over {t + 1}} = 4$;<br><br>$\Rightarrow$ t = 1, 3 = 5<sup>x $-$ 1</sup><br><br>$\Rightarrow$ x $-$ 1 = 0 (one of the possible value).<br><br>$\Rightarrow$ x = 1
About this question
Subject: Mathematics · Chapter: Binomial Theorem · Topic: General Term
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