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Let $g : R \to R$ be given by $g (x) = 3 + 4 x$ . If $g^{n} (x) = (gogo . .. og) (x)$ , and $g^{n} (x) = A + Bx$ then A and B are

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$2^{n + 1} - 1, 2^{n + 1}$

$4^{n} - 1, 4^{n}$

$3^{n}, 3^{n} + 1$

$5^{n} - 1, 5^{n}$

answer is B.

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Detailed Solution

Put n = 2, $g (g (x)) = g (3 + 4 x)$
$\begin{array}{l} 3 + 4 (3 + 4 x) g^{2} (x) = 15 + 16 x = (4^{2} - 1) + 4^{2} x \\ ∴ g^{n} (x) = (4^{n} - 1) + 4^{n} x \Rightarrow A = 4^{n} - 1; B = 4^{n} \end{array}$

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