Chapter 9, Problem 2PS

a.

To determine

Graph the first $10$ terms of the sequence.

Answer to Problem 2PS

  

Explanation of Solution

Calculation:

Let us find the first $10$ terms in the sequence

The sequence is $3+{\left(-1\right)}^n$

The first $10$ terms of the sequence given by $3+{\left(-1\right)}^n$ are

  $\begin{array}{l}{a}_1=3+{\left(-1\right)}^1\\ a_1=\mathrm{2.......}{\text{1}}^{\text{st}}\text{term}\\ a_2=3+{\left(-1\right)}^2\\ a_2=\mathrm{4.......}{\text{2}}^{\text{nd}}\text{term}\\ a_3=3+{\left(-1\right)}^3\\ a_3=\mathrm{2.......}{\text{3}}^{\text{rd}}\text{term}\\ a_4=3+{\left(-1\right)}^4\\ a_4=\mathrm{4.......}{\text{4}}^{\text{th}}\text{term}\\ a_5=3+{\left(-1\right)}^5\\ a_5=\mathrm{2.......}{\text{5}}^{\text{th}}\text{term}\end{array}$

  $\begin{array}{l}{a}_6=3+{\left(-1\right)}^6\\ a_6={\mathrm{4.......6}}^{th}\text{term}\\ a_7=3+{\left(-1\right)}^7\\ a_7={\mathrm{2.......7}}^{th}\text{term}\\ a_8=3+{\left(-1\right)}^8\\ a_8={\mathrm{4.......8}}^{th}\text{term}\\ a_9=3+{\left(-1\right)}^9\\ a_9={\mathrm{2.......9}}^{\text{th}}\text{term}\\ a_{10}=3+{\left(-1\right)}^{10}\\ a_{10}={\mathrm{4.......10}}^{\text{th}}\text{term}\end{array}$

Using the values derived above, we plot theses in the graph below

  

b.

To determine

Describe the behaviour of the graph of the sequence.

Answer to Problem 2PS

  $\text{odd }{a}_n=2,{\text{even a}}_n=4$

Explanation of Solution

Calculation:

Hence, the graph in (a) shows that when $n$ is odd $a_n=2$ and when $n$ is even, $a_n=4$

c.

To determine

Complete the table.

Answer to Problem 2PS

  $\begin{array}{cccccc}n& 1& 10& 101& 1000& 10,001\\ a_n& 2& 4& 2& 4& 2\end{array}$

Explanation of Solution

Calculation:

We put in different values of $n$ in the given equation We have seen in (b) that when $n$ is odd $a_n=2$ and when $n$ is even, $a_n=4$

Using the above, we can fill the table as below

Hence, we see that $1,101,10001$ are odd and $10$ and $1000$ are even

  $\begin{array}{cccccc}n& 1& 10& 101& 1000& 10,001\\ a_n& 2& 4& 2& 4& 2\end{array}$

d.

To determine

Determine (if possible) the value of $a_n$ as $n$ approaches infinity.

Answer to Problem 2PS

It is not possible to find the value of $a_n$ as $n$ approaches infinity

Explanation of Solution

Calculation:

We have seen in (b) that when $n$ is odd $a_n=2$ and when $n$ is even, $a_n=4$

This pattern continues as $n$ approaches infinity

Hence, it is not possible to find the value of $a_n,as,n$ approaches infinity.