2580.11 – Big f(x), Find f(6)

If f(n)f(n) is a function such that f(1)=f(2)=f(3)=1f(1) = f(2) = f(3) = 1, and is defined as

f(n+1)=f(n)⋅f(n−1)+1f(n−2) for n>3,f(n+1)=\frac{f(n)\cdot f(n-1)+1}{f(n-2)} \text{ for } n > 3,

then f(6)=f(6) =

  1. 2
  2. 3
  3. 7
  4. 11
  5. 26
Solution

We will compute f(4),f(5)f(4), f(5), and f(6)f(6):

f(4)=f(3+1)=f(3)⋅f(2)+1f(1)=1⋅1+11=2f(4)=f(3+1)=\frac{f(3)\cdot f(2)+1}{f(1)} =\frac{1\cdot1 + 1}{1} = 2

f(5)=f(4+1)=f(4)⋅f(3)+1f(2)=2⋅1+11=3f(5)=f(4+1)=\frac{f(4)\cdot f(3)+1}{f(2)} =\frac{2\cdot1 + 1}{1} = 3

f(6)=f(5+1)=f(5)⋅f4)+1f(3)=3⋅2+11=7f(6)=f(5+1)=\frac{f(5)\cdot f4)+1}{f(3)} =\frac{3\cdot2 + 1}{1} = 7

So the answer is (c).