Prove every third Fibonacci number is even.-- The Fibonacci numbers are the sequence 1,1,2,3,5,8,13,..., given by $\displaystyle f_n$ where $\displaystyle f$(1)=1, and $\displaystyle f$(2)=1, and $\displaystyle f_(n)=f(n-1)+(n-2)($ for all $\displaystyle n$.

This is the problem...

This is what I have...

Base Case:

f(3k-2) if k=1 then f(3(1)-2)=f(1)=1 odd #

f(3k-1) f(3(1)-1)=f(2)=1 odd #

f(3k) f(3(1) =f(3)=2 even #

so f(1)+f(2)=f(3)

an odd+odd=even

Inductive step:

Suppose it is true for k+1

f(3(k+1)-2)=f(3k+1)

f(3(k+1)-1)=f(3k+2)

f(3(k+1)) =f(3K+3)

I do not know where to go from here...