Question

Fibonacci Sequence In a study of the progeny of rabbits, Fibonacci (ca. \(1170-\) ca. 1240 ) encountered the sequence now bearing his name. It is defined recursively by \(a_{n+2}=a_{n}+a_{n+1}, \quad\) where \(\quad a_{1}=1\) 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 \(\rho=1+1 / \rho\) and solve this equation for \(\rho\).

Short answer

The first 12 terms of the Fibonacci sequence are: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144. The first 10 terms of the sequence \(b_{n}=\frac{a_{n+1}}{a_{n}}\) are approximately: 1, 2, 1.5, 1.6667, 1.6, 1.625, 1.6154, 1.6190, 1.6176, 1.6182. The golden ratio \(\rho = (1 + \sqrt{5}) / 2\) is approximately 1.61803.

Step-by-step solution

Construct the Fibonacci Sequence

Start with 1 and 1 as the first two terms. For each additional term, add the two preceding terms together. The first 12 terms of the Fibonacci sequence using the defined recursive function \(a_{n+2}=a_{n}+a_{n+1}\) are: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144.

Construct the Sequence \(b_{n}=\frac{a_{n+1}}{a_{n}}\)

The new sequence uses the values from the Fibonacci sequence. Use the first 11 terms of the Fibonacci sequence generated to create the first 10 terms of new sequence \(b_{n}\), they are: 1, 2, 1.5, 1.6667, 1.6, 1.625, 1.6154, 1.6190, 1.6176, 1.6182.

Prove the Relationship \(b_{n}=1+\frac{1}{b_{n-1}}\)

Rearrange the recursive Fibonacci formula: \(a_{n+2}=a_{n}+a_{n+1}\) in terms of \(a_{n+1}\) and \(a_{n}\) we get: \(a_{n+2}/a_{n+1} = a_{n}/a_{n+1} + 1\), which equates to \(b_{n+1} = b_{n}+1/b_{n}\).

Define and Solve for the Golden Ratio

The limit as \(n\) approaches infinity of the sequence \(b_n\) is defined as the golden ratio \(\rho = \lim_{n \to \infty} b_{n}\). Since the limit of \(b_n\) also satisfies the relationship \(b_n = 1 + 1/b_{n-1}\), we can substitute \(\rho\) to obtain \(\rho = 1 + 1 / \rho\), which simplifies to \(\rho² - \rho - 1 = 0\). Solving this quadratic equation gives the solutions \(\rho = (1 ± \sqrt{5}) / 2\). Since \(\rho\) is a ratio and cannot be negative, we reject the negative root, and thus \(\rho = (1 + \sqrt{5}) / 2\) which is approximately equal to 1.61803.