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WBJEE · Maths · Sequences and Series

If \(1, \log _9\left(3^{1-x}+2\right), \log _3\left(4.3^{\mathrm{x}}-1\right)\) are in A.P, then \(x\) equals

  1. A \(\log _3 4\)
  2. B \(1-\log _3 4\)
  3. C \(1-\log _4 3\)
  4. D \(\log _4 3\)
Verified Solution

Answer & Solution

Correct Answer

(B) \(1-\log _3 4\)

Step-by-step Solution

Detailed explanation

\begin{aligned} & \text {Hint: } 2 \log _9\left(3^{1-x}+2\right)=\log _3\left(4.3^x-1\right)+1 \\ & \Rightarrow \log _3\left(3^{1-x}+2\right)=\log _3 3\left(4.3^x-1\right) \\ & \Rightarrow 3^{1-x}+2=3\left(4.3^x-1\right) \\ & \Rightarrow \frac{3}{3^x}+2=4.3^{x+1}-3 \\ &…