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

JEE Main · 2023 · 30 Jan 2023 · Shift 1 · Sets Relations and Functions

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} \circ f^{n - 1} (x)

.
If

f^{5} (x) = \frac{a x + b}{b x + a}

g cd (a, b) = 1

, then

a + b

is equal to ____.

Answer: 3125

Represent the Mobius map

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

by the matrix

M = [3223]

, whose powers correspond to repeated composition. This symmetric form is preserved under multiplication, and with eigenvalues

5

and

1

M^{n} = [\frac{5 ^{n} + 1}{2} \frac{5 ^{n} - 1}{2} \frac{5 ^{n} - 1}{2} \frac{5 ^{n} + 1}{2}] .

For

n = 5

5^{5} = 3125

gives

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

, so with

g cd (a, b) = 1

a = 1563, b = 1562

and

a + b = 3125.

JEE Main · 2023 · 30 Jan 2023 · Shift 1 · Integer answer