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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?

Caleb Stanford
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Teru
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  • Another one here: http://math.stackexchange.com/questions/701888/proof-by-induction-that-if-n-in-mathbb-n-then-it-can-be-written-as-sum-of-di. – Martin R Mar 04 '16 at 06:02
  • Note that the question in the first duplicate suggested bu Martin R is not quite the same, but the accepted answer does also answer this question. – David Mar 04 '16 at 06:02

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