Fibonacci Sequence In a study of the progeny of rabbits,...

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 $$a_{n+2}=a_{n}+a_{n+1},$ where $a_{1}=1$ and $a_{2}=1 .$$ (a) Write the first 12 terms of the sequence. (b) Write the first 10 terms of the sequence defined by $$ b_{n}=\frac{a_{n+1}}{a_{n}}, \quad n \geq 1 $$ (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 $\lim _{n \rightarrow \infty} b_{n}=\rho$ . Show that $$\rho=1+\frac{1}{\rho}$$ and solve this equation for $\rho$

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
$$a_{n+2}=a_{n}+a_{n+1},$ where $a_{1}=1$ and $a_{2}=1 .$$
(a) Write the first 12 terms of the sequence.
(b) Write the first 10 terms of the sequence defined by
$$
b_{n}=\frac{a_{n+1}}{a_{n}}, \quad n \geq 1
$$
(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 $\lim _{n \rightarrow \infty} b_{n}=\rho$ . Show that
$$\rho=1+\frac{1}{\rho}$$
and solve this equation for $\rho$

Key Concepts

Recursive Sequences

Recursive sequences are defined by specifying one or more initial terms and a rule that computes each subsequent term as a function of the preceding ones. This method of definition is foundational in exploring patterns, convergence properties, and various applications across mathematics.

Fibonacci Sequence

The Fibonacci sequence is a famous example of a recursive sequence where each term is the sum of its two predecessors. It serves as a classical case study in recursion, combinatorics, and even appears in natural phenomena and art.

Ratio of Successive Terms

The ratio of successive terms in a recursive sequence, especially in the Fibonacci sequence, often leads to insights about the behavior of the sequence. Studying these ratios helps in understanding convergence properties and the emergence of limiting constants.

Limit and the Golden Ratio

As the Fibonacci sequence progresses, the ratio of consecutive terms converges to a constant known as the golden ratio. This convergence is significant because the golden ratio satisfies the self-referential equation ? = 1 + 1/?, which naturally leads to a quadratic equation whose positive solution has deep mathematical and aesthetic implications.