Hello,
you can use the fact that and . This will allow you to set up an induction step.
The Fibonacci Numbers : 1,1,2,3,5,8,13,21,34... are defined by and for n >2 by .
Show by induction that is even.
base case: n=3 is 2, so it is even.
Assume is even?
not sure how to do the induction step.
Thanks in advance
The Fibonacci sequence, in terms of odd and even is...
OOEOOEOOEOOEOOEOOE.....
This is because the first 2 terms are odd
odd+odd=even=3rd term
odd+even=odd=4th term
even+odd=odd=5th term
odd+odd=even=6th term
The sequence cycles in this way
Hence if
The inductive step is..... does being even cause to be even ?
Proof
If is even, then both and must of necessity both be odd.
Hence, as this is true, and is even, is even.
The inductive proof is really just basic logic in this case!