Chapter 9.1, Problem 83E

Fibonacci Sequence In a study of the progeny of rabbits, Fibonacci (ca. 1170-ca. 1240) encountered the sequence now bearing his name. The sequence is defined recursively as $a_{n+2}=a_n+a_{n+1}$ where $a_1=1$ and $a_1=2$

(a) Write the first 12 terms of the sequence.

(b) Write the first 10 terms of the sequence defined by $b_n=\frac{a_{n-1}}{a_n},n\ge 1$

(c) Using the definition in part (b), show that $b_n=1+\frac{1}{b_{n-1}}$.

(d) The golden ratio $\rho$ can be defined by $\underset{n-\infty }{\lim }{b}_n=\rho$ Show that $\rho =1+(1/\rho )$ and solve this equation for $\rho$.