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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 , gcd a , b = 1 , then a + b is equal to ________

Correct answer

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Step-by-step solution

Given: f 1 x = 3 x + 2 2 x + 3 Now, f 2 x = f 1 o f 1 x ⇒ f 2 x = 3 3 x + 2 2 x + 3 + 2 2 3 x + 2 2 x + 3 + 3 ⇒ f 2 x = 9 x + 6 + 4 x + 6 6 x + 4 + 6 x + 9 ⇒ f 2 ( x ) = 13 x + 12 12 x + 13 ⇒ f 3 x = 3 13 x + 12 12 x + 13 + 2 2 13 x + 12 12 x + 13 + 3 ⇒ f 3 x = 63 x + 62 62 x + 63 ⇒ f 4 x = 3 63 x + 62 62 x + 63 + 2 2 63 x + 62 62 x + 63 + 3 ⇒ f 4 x = 313 x + 312 312 x + 313 ∴ f 5 x = 1563 x + 1562 1562 x + 1563 = a x + b b x + a ∴ a + b = 1563 + 1562 = 3125

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