2615.11 – A Mystery Function

If f(x)=4x/f(x+1)f(x) = 4x/f(x+1) and f(2)=1/3f(2) = 1/3, then what is f(6)f(6)?

  1. 5/85/8
  2. 15/415/4
  3. 32/532/5
  4. 88
  5. 48/548/5
Solution

The given formula for ff leads to a more useful form,

f(x)=4xf(x+1)⇝f(x+1)=4xf(x).f(x) = \frac{4x}{f(x+1)} \leadsto f(x+1) = \frac{4x}{f(x)}.

The latter form is more useful because it can be used inductively, to find a value of ff at a larger number from its value at a smaller number. Starting with the given value at 22 it will be just four steps to 66.

f(2)=13⇝f(3)=f(2+1)=81/3=24⇝f(4)=f(3+1)=12/24=1/2⇝f(5)=161/2=32⇝f(6)=2032=58.\begin{aligned} f(2) = \frac{1}{3} &\leadsto& f(3) = f(2 + 1) = \frac{8}{1/3} = 24 \\ &\leadsto& f(4) = f(3+1) = 12/24 = 1/2 \\ &\leadsto& f(5) = \frac{16}{1/2} = 32 \\ &\leadsto& f(6) = \frac{20}{32} = \frac{5}{8}. \end{aligned}

The answer is a.