First slide

Binomial theorem for positive integral Index

Question

The value of x, for which the 6th term in the expansion of ${[2^{\log_{2} \sqrt{(9^{x - 1} + 7)}} + \frac{1}{2^{\frac{1}{5} \log_{2} (3^{x - 1} + 1)}}]}^{7}$ is 84, is equal to

Moderate

A

4
B

3
C

2
D

1

Solution

The given expression
$\begin{aligned} = {[\sqrt{9^{x - 1} + 7} + \frac{1}{{(3^{x - 1} + 1)}^{1 / 5}}]}^{7} \\ Given, T_{6} = 84 \\ \Rightarrow^{7} C_{5} {(\sqrt{9^{x - 1} + 7})}^{2} {(\frac{1}{{(3^{x - 1} + 1)}^{1 / 5}})}^{5} = 84 \\ \Rightarrow^{7} C_{5} (9^{x - 1} + 7) \cdot \frac{1}{(3^{x - 1} + 1)} = 84 \\ \Rightarrow 9^{x - 1} + 7 = 4 (3^{x - 1} + 1) \\ \Rightarrow 3^{2 x} - 12 \cdot 3^{x} + 27 = 0 \\ \Rightarrow (3^{x} - 3) (3^{x} - 9) = 0 \\ \to 3^{x} = 3, 9 \to x = 1, 2 \end{aligned}$

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