Prove by induction that the -th Fibonacci number satisfies the equality

where is the golden ratio and is its conjugate.

From chapter text, the values of and are as follows:

And, Fibonacci series is defined by:


So, the equality holds for the basis steps.

Inductive Step: Let us assume that the inequality holds for and such that . We have to show that it holds for too.

So, the inequality holds for also. Hence, the proof is complete.

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