Kenneth Rosen Edition 7 Exercise 8.2 Question 41 (Page No. 526)

Question

• A. Use the formula found in Example $4$ for $f_{n},$ the $n^{\text{th}}$ Fibonacci number, to show that fn is the integer closest to $$\dfrac{1}{\sqrt{5}}\left(\dfrac{1 + \sqrt{5}}{2}\right)^{n}$$
• B. Determine for which $n\: f_{n}$ is greater than $$\dfrac{1}{\sqrt{5}}\left(\dfrac{1 + \sqrt{5}}{2}\right)^{n}$$ and for which $n\: f_{n}$ is less than $$\dfrac{1}{\sqrt{5}}\left(\dfrac{1 + \sqrt{5}}{2}\right)^{n}.$$