Chapter 6.6, Problem 54E

(a)

To determine

To write sequence in which each term $a_n$ is the side length of square n

Answer to Problem 54E

Sequence in which each term $a_n$ is the side length of square n is

  $1,1,2,3,5,\mathrm{8....}$

Explanation of Solution

Given:

Squares in following diagram

  

Concept Used:

Fibonacci series: A series of numbers in which each number (Fibonacci number) is the sum of the two preceding numbers.

Calculation:

From the given figure taking length of square sides in the order form a sequence of squares

  $1,1,2,3,5,\mathrm{8....}$

Conclusion:

Sequence in which each term $a_n$ is the side length of square n is

  $1,1,2,3,5,\mathrm{8....}$

(b)

To determine

To write name of sequence and determine next term of sequence

Answer to Problem 54E

Sequence obtained in part (a) is

As

  $\begin{array}{l}1+1=2\\ 2+1=3\\ 3+2=5\\ 5+3=8\\ 8+5=13\end{array}$

Therefore,

These are Fibonacci numbers

Therefore recursive rule is given by

  $a_n=a_{n-2}+a_{n-1}$ With $\left[a\left(1\right)=1,a\left(2\right)=1,n>2\right]$

Name of the sequence is Fibonacci sequence

Next term of the sequence is 13

Explanation of Solution

Given:

Squares in following diagram

  

Concept Used:

Fibonacci series: a series of numbers in which each number (Fibonacci number) is the sum of the two preceding numbers.

Calculation:

Sequence obtained is $1,1,2,3,5,\mathrm{8....}$

This sequence satisfies condition of Fibonacci series

Hence,

Given sequence is Fibonacci sequence

The next term of this sequence is sum of previous two

Therefore,

Next term = 8+5 = 13

Conclusion:

Name of the sequence is Fibonacci sequence

Next term of the sequence is 13

(c)

To determine

To add another square in the given diagram

Answer to Problem 54E

  

Explanation of Solution

Given:

Squares in following diagram

  

Concept Used:

Using next term obtained in part (b)

Calculation:

From the sequence $1,1,2,3,5,8,\mathrm{13....}$

Next Square to drawn will have 13 as its side

Therefore,

Diagram obtained is given below