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.