The Fibonacci numbers Fn are defined by the recurrence Fn=Fn−1+Fn−2, with base cases F0=0 and F1=1.
Prove that any non-negative integer can be written as the sum of distinct and non-consecutive Fibonacci numbers. That is, if FiFi appears in the sum, then Fi−1 and Fi+1 do not appear in the sum. For example:
17=F7+F4+F2
42=F9+F6
54=F9+F7+F5+F3+F1
How can i prove this?