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

Correct Answer: 3125

The function \( f_1(x) \) is given as:

\[ f_1(x) = \frac{3x + 2}{2x + 3} \]

First Iteration:

\[ f_1(x) = \frac{3x + 2}{2x + 3} \]

Second Iteration:

\[ f_2(x) = f_1(f_1(x)) = \frac{13x + 12}{12x + 13} \]

Notice how the numerator and denominator coefficients evolve as the function is iterated.

Third Iteration:

\[ f_3(x) = f_1(f_2(x)) = \frac{63x + 62}{62x + 63} \]

The pattern becomes clearer as we proceed further. Observe the symmetry in the coefficients.

Fifth Iteration:

\[ f_5(x) = \frac{1563x + 1562}{1562x + 1563} \]

This results from applying the function iteratively, maintaining the structure of coefficients in the numerator and denominator.

Given Condition:

The condition provided is:

\[ a + b = 3125 \]

Here, \( a \) and \( b \) are the coefficients of \( x \) and the constant term in the numerator of \( f_5(x) \), respectively.

Conclusion:

We have \( a = 1563 \) and \( b = 1562 \), so:

\[ a + b = 1563 + 1562 = 3125 \]

Thus, the given condition is satisfied.

Answer 2

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