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. 58\dfrac{5}{8}

  2. 154\dfrac{15}{4}

  3. 325\dfrac{32}{5}

  4. 88

  5. 485\dfrac{48}{5}

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Solution

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

f(x)=4xf(x+1)⇝f(x+1)=4xf(x)f(x) = \dfrac{4x}{f(x+1)} \leadsto f(x+1) = \dfrac{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) = \dfrac{1}{3} &\leadsto f(3) = f(2 + 1) = \dfrac{8}{1/3} = 24 \\&\leadsto f(4) = f(3+1) = 12/24 = 1/2 \\&\leadsto f(5) = \dfrac{16}{1/2} = 32 \\&\leadsto f(6) = \dfrac{20}{32} = \dfrac{5}{8}\end{aligned}

The answer is a.