If m, element of N, is a multiple of 4 then Fm is a multiple of 3.

Question

The Fibonacci numbers are denoted to be F1 = 1, F2 = 1, and Fn = Fn-1 + Fn-2 for n > 2.

 

<Mathematical Proof question>

Answer 1

The series is 1 1 2 3 5 8 13 21 etc.

We can see that every 4th term is a multiple of 3: 3 21, etc so we need to look at the series closely to find out why. F2=F1=1. F4=F3+F2=F2+F1+F2=3F1

F8=F7+F6=F6+F5+F5+F4=F6+2F5+F4=F5+F4+2F5+F4=3F5+2F4,

But we know F4 is a multiple of 3 (F4=3F1), so F8=3F5+6F1=3(F5+2F1),

Therefore F8 is also a multiple of 3.

By induction, every 4th term is a multiple of 3, so Fm is a multiple of 3 when m is a natural number which is a multiple of 4.