Chapter 5.1, Problem 3SP

The Fibonacci numbers are defined recursively by the sequence {F_n} where F₀= 0. F₁= I and for is $n\ge 2$ . $F_n=F_{n-1}+F_{n-2}$Here we look at properties of the Fibonacci numbers.

3. Use the answer in 2c. to show that

   $\underset{n\to \infty }{\lim }\frac{{}^{F}n+1}{F_n}=\frac{1+\sqrt{5}}{2}.$

The number $\varphi =\left(1+\sqrt{5}\right)/2$ is known as the golden ratio (Figure 5.8 and Figure 5.9)

Figure 5.8 The seeds in a sunflower exhibit spiral patterns curving to the left and to the right. The number of spirals in each direction is always a Fibonacci number—.always. (credit: modification of work by Esdras Calderan, Wikixnedia Commons)

Figure 5.9 The propoition of the golden ratio appears in many famous examples of art and architecture. The ancient Greek temple known as the Panhenon was designed s1th these proportions, and the ratio appears again in many of the smaller details. (credit: modification of bodc by TravelingOner, Flickr)