Let f1(x)=(3 x+2/2 x+3), x ∈ R - (-3/2) For n ≥ 2, define f n (x)=f1 o f n -1(x). If f5(x)=( a x+ b / b x+ a ), operatornamegcd( a , b )=1, then a + b is equal to

Question

Let f1(x)=(3 x+2/2 x+3), x ∈ R - (-3/2) For n ≥ 2, define f n (x)=f1 o f n -1(x). If f5(x)=( a x+ b / b x+ a ), operatornamegcd( a , b )=1, then a + b is equal to  (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

f1(x)=(3 x+2/2 x+3) ⇒ f2(x)=(13 x+12/12 x+13) ⇒ f3(x)=(63 x+62/62 x+63) ∴ f5(x)=(1563 x+1562/1562 x+1563) a+b=3125

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 $f^{5} (x) = \frac{a x + b}{b x + a}, gcd (a, b) = 1$ , then $a + b$ is equal to ______

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

Q. Let f1(x)=2x+33x+2​,x∈R−{2−3​} For n≥2, define fn(x)=f1ofn−1(x). If f5(x)=bx+aax+b​,gcd(a,b)=1, then a+b is equal to ______

f1(x)=2x+33x+2​ ⇒f2(x)=12x+1313x+12​ ⇒f3(x)=62x+6363x+62​ ∴f5(x)=1562x+15631563x+1562​ a+b=3125

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 $f^{5} (x) = \frac{a x + b}{b x + a}, gcd (a, b) = 1$ , then $a + b$ is equal to ______

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