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Question
If x = (43) (46) (46) (49) .... (43x) = (0.0625)−54, the value of x is
Options
(a) 7
(b) 8
(c) 9
(d) 10
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Solution
(b) 8
\[\left( 4^3 \right)\left( 4^6 \right)\left( 4^9 \right)\left( 4^{12} \right) . . . \left( 4^{3x} \right) = \left( 0 . 0625 \right)^{- 54} \]
\[ \Rightarrow 4^\left( 3 + 6 + 9 + 12 + . . . + 3x \right) = \left( \frac{625}{10000} \right)^{- 54} \]
\[ \Rightarrow 4^{3\left( 1 + 2 + 3 + 4 + . . . + x \right)} = \left( \frac{1}{16} \right)^{- 54} \]
\[ \Rightarrow 4^{3\left( \frac{x\left( x + 1 \right)}{2} \right)} = \left( \frac{1}{16} \right)^{- 54} \]
\[ \Rightarrow 4^{3\left( \frac{x\left( x + 1 \right)}{2} \right)} = \left( 4^{- 2} \right)^{- 54} \]
\[\text{ Comparing both the sides }: \]
\[ \Rightarrow 3\left( \frac{x\left( x + 1 \right)}{2} \right) = 108\]
\[ \Rightarrow x\left( x + 1 \right) = 72\]
\[ \Rightarrow x^2 + x - 72 = 0\]
\[ \Rightarrow x^2 + 9x - 8x - 72 = 0\]
\[ \Rightarrow x\left( x + 9 \right) - 8\left( x + 9 \right) = 0\]
\[ \Rightarrow \left( x + 9 \right)\left( x - 8 \right) = 0\]
\[ \Rightarrow x = 8, - 9\]
\[ \Rightarrow x = 8 [ \because \text{ x is positive }]\]
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