Chapter 12.1, Problem 92AYU

Droste Effect The Droste Effect, named after the image on boxes of Droste cocoa powder, refers to an image that contains within it a smaller version of the image. which in turn contains an even smaller version, and so on.

List the first 11 terms of the Fibonacci sequence.

List the first 10 terms of the ratio $\frac{u_{n+1}}{u_n}$

As it gets large. what number does the ratio approach? This number is referred to as the golden ratio. Rectangles whose sides are in this ratio were considered pleasing to the eye by the Greeks. For example, the facade of the Parthenon was constructed using the golden ratio.

Write down the first $10$ terms of the ratio $\frac{u_n}{u_{n+1}}$

As $n$ gets large, what number does the ratio approach? This number is referred to as the conjugate golden ratio. This ratio is believed to have been used in the construction of the Great Pyramid in Egypt. The ratio equals the sum of the areas of the four face triangles divided by the total surface area of the Great Pyramid.

Fibonacci Sequence Let

$u_n=\frac{{\left(1+\sqrt{5}\right)}^n-{\left(1-\sqrt{5}\right)}^n}{2^n\sqrt{5}}$

Define the $n\text{th}$ term of a sequence.

Show that $u_1=1$ and $u_2=1$.

Show that $u_{n+2}=u_{n+1}+u_n$.

Draw the conclusion that $\left\{u_n\right\}$ is a Fibonacci sequence.