Chapter 9.3, Problem 55E

Solution

To identify: The first seven terms of the “geometric sequence” with the first term the matrix $\left[11\right]$ and the common ratio the matrix $\left[\begin{array}{l}11\\ 01\end{array}\right]$ . How is this sequence of matrices related to the Fibonacci sequence?

The Fibonacci’s sequence $b_n$ is $1,1,2,3,5,8,13,21,\mathrm{...}$ .

Given information:

The given term is

  $\begin{array}{c}{a}_1=\left[11\right]\\ r=\left[\begin{array}{l}11\\ 01\end{array}\right]\end{array}$

Explanation:

Consider the given information.

Determine the value of $a_2$ by $a_{1}r$ .

  $\begin{array}{c}{a}_2=a_{1}r\\ a_2=\left[11\right]\cdot \left[\begin{array}{l}11\\ 01\end{array}\right]\\ a_2=\left[1\left(0\right)+1\left(1\right)\cdot 1\left(1\right)+1\left(1\right)\right]\\ a_2=\left[12\right]\end{array}$

Determine the value of $a_3$ by $a_{2}r$ .

  $\begin{array}{c}{a}_3=a_{2}r\\ a_3=\left[12\right]\cdot \left[\begin{array}{l}01\\ 11\end{array}\right]\\ a_3=\left[1\left(0\right)+2\left(1\right)\cdot 1\left(1\right)+2\left(1\right)\right]\\ a_3=\left[23\right]\end{array}$

Determine the value of $a_4$ by $a_{3}r$.

  $\begin{array}{c}{a}_4=a_{3}r\\ a_4=\left[23\right]\cdot \left[\begin{array}{l}01\\ 11\end{array}\right]\\ a_4=\left[2\left(0\right)+3\left(1\right)\cdot 2\left(1\right)+3\left(1\right)\right]\\ a_4=\left[35\right]\end{array}$

Determine the value of $a_5$ by $a_{4}r$

  $\begin{array}{c}{a}_5=a_{4}r\\ a_5=\left[35\right]\cdot \left[\begin{array}{l}01\\ 11\end{array}\right]\\ a_5=\left[3\left(0\right)+5\left(1\right)\cdot 3\left(1\right)+5\left(1\right)\right]\\ a_5=\left[58\right]\end{array}$

Determine the value of $a_6$ by $a_{5}r$

  $\begin{array}{c}{a}_6=a_{5}r\\ a_6=\left[58\right]\cdot \left[\begin{array}{l}01\\ 11\end{array}\right]\\ a_6=\left[5\left(0\right)+8\left(1\right)\cdot 5\left(1\right)+8\left(1\right)\right]\\ a_6=\left[813\right]\end{array}$

Determine the value of $a_7$ by $a_{6}r$

  $\begin{array}{c}{a}_7=a_{6}r\\ a_7=\left[813\right]\cdot \left[\begin{array}{l}01\\ 11\end{array}\right]\\ a_7=\left[8\left(0\right)+13\left(1\right)\cdot 8\left(1\right)+13\left(1\right)\right]\\ a_7=\left[1321\right]\end{array}$

Each of the matrices in the sequence $a_n$ has as element two consecutive terms of the Fibonacci’s sequence.

  $a_k=\left[b_{k}b\right]$

The Fibonacci’s sequence $b_n$ is $1,1,2,3,5,8,13,21,\mathrm{...}$ .