Chapter 13, Problem 60E

a. Explain what happens to the values of ${\left(\frac{1-\sqrt{5}}{2}\right)}^N$ as $N$ gets larger. (Hint: Get a calculator and experiment with $N=6,7,8,\mathrm{....}$ until you get the picture.)

b. Explain why $F_N\to \frac{{\varphi }^N}{\sqrt{5}}$. [Hint: Use (a) and the original Binet’s formula]

c. Using (b), explain why $\left(\frac{F_{N+1}}{F_N}\right)\to \varphi$.