Chapter 7.10, Problem 30PPS

a.

To determine

To calculate: The relationship between the terms of the sequence and to calculate the next five terms in the sequence.

Answer to Problem 30PPS

The next five terms are 13,21,34,55,89.

Explanation of Solution

Given:

The first term are 1,1,2,3,5,8,….

Calculation:

Notice that for the sequence 1,1,2,3,4,5,8…….

  $\begin{array}{l}{a}_1+a_2=1+1=2=a_3.\\ a_2+a_3=1+2=3=a_4.\\ a_3+a_4=2+3=5=a_5.\\ a_4+a_5=3+5=8=a_6.\end{array}$

Each term is then the sum of the previous two terms. The next five terms are then:

  $\begin{array}{l}{a}_7=a_5+a_6=5+8=13.\\ a_8=a_6+a_7=8+13=21.\\ a_9=a_7+a_8=13+21=34.\\ a_{10}=a_8+a_9=21+34=55.\\ a_{11}=a_9+a_{10}=34+55=89.\\ \end{array}$

So, next five terms are 13,21,34,55,89.

b.

To determine

To write: The formula for the nth term if $a_1=1,a_2=1\text{and}n\ge 3.$

Answer to Problem 30PPS

  $a_1=1,a_2=1,a_n\text{=}{a}_{n-2}+a_{n-1}\text{ ,}n\ge 3.$

Explanation of Solution

Given:

The first term are 1,1,2,3,5,8,…

Calculation:

From part (a), each term is the sum of the previous two term so,

  $a_n=a_{n-2}+a_{n-1}$ . The formula is then,

  $a_1=1,a_2=1,a_n\text{=}{a}_{n-2}+a_{n-1}\text{ ,}n\ge 3.$

c.

To determine

To find: The 15^th term of the series.

Answer to Problem 30PPS

The 15^th term is 610.

Explanation of Solution

Given:

The first term are 1,1,2,3,5,8,….

Calculation:

Using the recursive formula,

  $a_1=1,a_2=1,a_n\text{=}{a}_{n-2}+a_{n-1}\text{ ,}n\ge 3.$

  $\begin{array}{l}{a}_7=a_5+a_6=5+8=13.\\ a_8=a_6+a_7=8+13=21.\\ a_9=a_7+a_8=13+21=34.\\ a_{10}=a_8+a_9=21+34=55.\\ a_{11}=a_9+a_{10}=34+55=89.\\ a_{12}=a_{10}+a_{11}=55+89=144.\\ a_{13}=a_{11}+a_{12}=89+144=233.\\ a_{14}=a_{12}+a_{13}=144+233=377.\\ a_{15}=a_{13}+a_{14}=233+377=610.\end{array}$

The 15^th term is 610.

Conclusion:

d.

To determine

To Explain: The reason that Fibonacci sequence is not an arithmetic sequence.

Answer to Problem 30PPS

It has no common difference.

Explanation of Solution

Given:

The first term are 1,1,2,3,5,8,….

Calculation:

Fibonacci sequence is not an arithmetic sequence because it has no common difference.