Let f 1 ( x ) = 3 x + 2 2 x + 3 , x R - { - 3 2 } For n 2 , define f n ( x ) = f 1 o f n - 1 ( x ) . If f 5 ( x ) = a x + b b x + a , g c d ( a , b ) = 1 , then a + b is equal to _________ . [2023]

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Solution

Q.
Let $f^{1} (x) = \frac{3 x + 2}{2 x + 3}, x \in R - {\frac{- 3}{2}}$

For $n \geq 2,$ define $f^{n} (x) = f^{1} o f^{n - 1} (x) .$

If $f^{5} (x) = \frac{a x + b}{b x + a}, g c d (a, b) = 1,$ then a + b is equal to _________ . [2023]

Ans.
(3125)

We have, $f^{1} (x) = \frac{3 x + 2}{2 x + 3}, x \in ℝ - {\frac{- 2}{3}}$

$f^{2} (x) = \frac{13 x + 12}{12 x + 13}; f^{3} (x) = \frac{63 x + 62}{62 x + 63}$

$∴$ $f^{5} (x) = \frac{1563 x + 1562}{1562 x + 1563} = \frac{a x + b}{b x + a}$

$∴$ $a + b = 1563 + 1562 = 3125$

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