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

where $\phi$ is the golden ratio and $\hat\phi$ is its conjugate.

From chapter text, the values of $\phi$ and $\hat\phi$ are as follows:

And, Fibonacci series is defined by:

Basis:

So, the equality holds for the basis steps.

Inductive Step: Let us assume that the inequality holds for $i = k$ and $i = k - 1$ such that $k \ge 2$. We have to show that it holds for $i = k + 1$ too.

So, the inequality holds for $k + 1$ also. Hence, the proof is complete.