Let f (x)=(x+1/x-1) for all x ≠1 Let f1(x)=f (x), f2(x) =f (f(x)) and generally fn (x) =f (fn-1(x)) for n 1 Let P=f1(2)f2(3)f3(4)f4(5). Which of the following is a multiple of P ?

Question

Let f (x)=(x+1/x-1) for all x ≠1 Let f1(x)=f (x), f2(x) =f (f(x)) and generally fn (x) =f (fn-1(x)) for n > 1 Let P=f1(2)f2(3)f3(4)f4(5). Which of the following is a multiple of P ? (A) 125 (B) 375 (C) 250 (D) 147

Tardigrade Admin · Accepted Answer

We have, f1 (x) =(x+1/x-1) f2(x)=f (f (x))=(f (x)+1/f (x) -1)=((x+1/x-1)+1/(x+1)x-1-1)=x f3 = f (x), f4 (x) =x P=f1(2). f2(3). f3(4). f4 (5) P=3× 3× (5/3)× 5 =75 ∴ Multiple of P is 375

Q. Let f(x)=x−1x+1​ for all x=1 Let f1(x)=f(x),f2(x)=f(f(x)) and generally fn(x)=f(fn−1(x)) for n>1 Let P=f1(2)f2(3)f3(4)f4(5). Which of the following is a multiple of P ?

125

375

250

147

We have, f1(x)=x−1x+1​ f2(x)=f(f(x))=f(x)−1f(x)+1​=x−1x+1​−1x−1x+1​+1​=x f3=f(x),f4(x)=x P=f1(2).f2(3).f3(4).f4(5) P=3×3×35​×5=75 ∴ Multiple of P is 375

Q. Let $f (x) = \frac{x + 1}{x - 1}$ for all $x \neq = 1$ 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 $P$ ?