Question

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

Answer 1

$f^1(x)=\frac{3 x+2}{2 x+3}$
$\Rightarrow f^2(x)=\frac{13 x+12}{12 x+13}$
$\Rightarrow f^3(x)=\frac{63 x+62}{62 x+63}$
$\therefore f^5(x)=\frac{1563 x+1562}{1562 x+1563}$
$a+b=3125$