For let be the Fibonacci number :
Prove that :
According to...
Fibonacci number - Wikipedia, the free encyclopedia
... a basic property of the fibonacci's numbers is that 'every 3rd number of the sequence is even and more generally, every k-th number of the sequence is a multiple of '. In case of is ...
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Proof by induction:
Let the statement be P(n).
Thus P(3) is true.
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Let P(m) be true.
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Consider P(m+3).
Thus 3 divides m+3.
Thus is even.
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Hence, by the principle of induction, the statement is true.