Chapter 12.7, Problem 40E

a.

To determine

Find $\frac{F_n}{F_{n-1}}$ for second to eleven term of Fibonacci sequence.

Answer to Problem 40E

  $\begin{array}{cccccccccccc}Term& 1& 2& 3& 4& 5& 6& 7& 8& 9& 10& 11\\ Ratio& & 1& 2& 1.5& 1.667& 1.6& 1.625& 1.615& 1.619& 1.618& 1.618\end{array}$

Explanation of Solution

Given information:

Fibonacci sequence $1,1,2,3,\mathrm{....},F_{n-2}+F_{n-1}$

Calculation:

Consider the given sequence,

  $1,1,2,3,\mathrm{....},F_{n-2}+F_{n-1}$

Now calculate the $\frac{F_n}{F_{n-1}}$ , for second to eleven term of Fibonacci sequence.

  $\begin{array}{cccccccccccc}Term& 1& 2& 3& 4& 5& 6& 7& 8& 9& 10& 11\\ Ratio& & 1& 2& 1.5& 1.667& 1.6& 1.625& 1.615& 1.619& 1.618& 1.618\end{array}$

b.

To determine

Identify the sequence of ratios are arithmetic, geometric or nothing.

Answer to Problem 40E

Nothing.

Explanation of Solution

Given information:

Fibonacci sequence $1,1,2,3,\mathrm{....},F_{n-2}+F_{n-1}$

Calculation:

Consider the ratio from table,

  $\begin{array}{cccccccccccc}Term& 1& 2& 3& 4& 5& 6& 7& 8& 9& 10& 11\\ Ratio& & 1& 2& 1.5& 1.667& 1.6& 1.625& 1.615& 1.619& 1.618& 1.618\end{array}$

The sequence of ratios is not arithmetic nor geometric because there is no common difference.

c.

To determine

Plot the graph of terms found in part $\left(a\right)$ .

Answer to Problem 40E

  

Explanation of Solution

Given information:

Fibonacci sequence $1,1,2,3,\mathrm{....},F_{n-2}+F_{n-1}$

Calculation:

The graph is as shown,

  

d.

To determine

Find the limit of sequence based on the graph.

Answer to Problem 40E

  $1.618$

Explanation of Solution

Given information:

Fibonacci sequence $1,1,2,3,\mathrm{....},F_{n-2}+F_{n-1}$

Calculation:

The limit of the sequence is, $1.618$

e.

To determine

Compare the value of golden ratio $1.61804$ with limit of sequence.

Answer to Problem 40E

The limit of the sequence is equal to golden ratio.

Explanation of Solution

Given information:

Fibonacci sequence $1,1,2,3,\mathrm{....},F_{n-2}+F_{n-1}$

Calculation:

The value of golden ratio and limit of the sequence is, approximately same.

The limit of the sequence is equal to golden ratio.

f.

To determine

Research the term golden ratio.

Answer to Problem 40E

The term golden ratio is the limit of the ratio successive terms of the Fibonacci sequence.

Explanation of Solution

Given information:

Fibonacci sequence $1,1,2,3,\mathrm{....},F_{n-2}+F_{n-1}$

Calculation:

If the wall hanging is in rectangular shape and having length and breadth ratio is equal to $1.618$ then eyes feel ease to see it

The term golden ratio is the limit of the ratio successive terms of the Fibonacci sequence.