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 a₁. = 1 and a₂ = 1.

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

(b) Write the first 10 terms of the sequence defined by

$b_n=\frac{a_n+1}{a_n}\text{, }n\ge 1.$

(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 $\underset{n\to \infty }{\lim }{b}_n=\rho .$ Show that $\rho =1+(1/\rho )$ and solve this equation for $\rho$.