Question

Let $f \left(x\right)=\frac{x+1}{x-1}$ for all $x \ne1$ Let $f^{1}(x)=f (x), f^{2}(x) =f (f(x))$ and generally $f^{n} (x) =f (f^{n-1}(x))$ for $n >\,1$ Let $P=f^{1}(2)f^{2}(3)f^{3}(4)f^{4}(5)$.
Which of the following is a multiple of $P$ ?

• A. 125
• B. 375
• C. 250
• D. 147

Answer 1

Answer: (B)

We have, $f^{1} (x) =\frac{x+1}{x-1}$
$f^{2}(x)=f (f (x))=\frac{f (x)+1}{f (x) -1}=\frac{\frac{x+1}{x-1}+1}{\frac{x+1}{x-1}-1}=x$
$f^{3} = f (x), f^{4} (x) =x$
$P=f^{1}(2). f^{2}(3). f^{3}(4). f^{4} (5)$
$P=3\times 3\times \frac{5}{3}\times 5 =75$
$\therefore $ Multiple of $P$ is $375$