Let f ^1( x )=frac{3 x +2}{2 x +3}, x ∈ R -left{frac{-3}{2}right} For n ≥ 2 , define f ^{ n }( x

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1.Relation and Function

hard

Let $f ^1( x )=\frac{3 x +2}{2 x +3}, x \in R -\left\{\frac{-3}{2}\right\}$ For $n \geq 2$, define $f ^{ n }( x )= f ^1 0 f ^{ n -1}( x )$. If $f ^5( x )=\frac{ ax + b }{ bx + a }, \operatorname{gcd}( a , b )=1$, then $a + b$ is equal to $............$.

A

$3124$

B

$3123$

C

$3126$

D

$3125$

(JEE MAIN-2023)

Solution

$f^1(x)=\frac{3 x+2}{2 x+3}$

$\Rightarrow f^2(x)=\frac{13 x+12}{12 x+13}$

$\Rightarrow f^3(x)=\frac{63 x+62}{62 x+63}$

$\therefore f^5(x)=\frac{1563 x+1562}{1562 x+1563}$

$a+b=3125$

Standard 12

Mathematics

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