Chapter 3.6, Problem 24PPS

a)

To determine

To list: first five terms of the arithmetic sequence given by the graph.

Answer to Problem 24PPS

The first five terms of the arithmetic sequence are $-3$ , $-1$ , 1, 3, and 5.

Explanation of Solution

Given:

A graph is shown below:

Calculation

Consider the given graph.

  

The points on the graph are $\left(1,-3\right),\left(2,-1\right),\left(3,1\right),\left(4,3\right),\left(5,5\right)$ . Because the terms are the second coordinate in each coordinate pair, the terms of the arithmetic sequence are $-3$ , $-1$ , 1, 3, and 5.

b)

To determine

To write: a formula to represent the nth term of the sequence.

Answer to Problem 24PPS

A formula to represent the nth term of the sequence is $a_n=2n-5$ .

Explanation of Solution

Given:

A graph is shown below:

Calculation

Consider the terms.

  $-3$ , $-1$ , 1, 3, and 5

Now, find the common difference of the sequence as shown:

  $-1+3=1+1=3-1=5-3=2$

The sequence is increasing, so the common difference is positive: 2.

Write the equation for the nth term of an arithmetic sequence using the first term $-3$ and common difference 2.

  $\begin{array}{l}{a}_n=a_1+\left(n-1\right)d\\ a_n=-3+\left(n-1\right)2\\ a_n=-3+2n-2\\ a_n=2n-5\end{array}$

So, formula to represent the nth term of the sequence is $a_n=2n-5$ .

c)

To determine

To write: a function to represent the nth term of the sequence.

Answer to Problem 24PPS

A function to represent the nth term of the sequence is $f\left(n\right)=2n-5$ .

Explanation of Solution

Given:

A graph is shown below:

Calculation

Consider the terms.

  $-3$ , $-1$ , 1, 3, and 5

Now, find the common difference of the sequence as shown:

  $-1+3=1+1=3-1=5-3=2$

The sequence is increasing, so the common difference is positive: 2.

Write the equation for the nth term of an arithmetic sequence using the first term $-3$ and common difference 2.

  $\begin{array}{l}{a}_n=a_1+\left(n-1\right)d\\ a_n=-3+\left(n-1\right)2\\ a_n=-3+2n-2\\ a_n=2n-5\end{array}$

So, formula to represent the nth term of the sequence is $a_n=2n-5$ .

So, the required function is $f\left(n\right)=2n-5$

d)

To determine

To find: the domain and range of the function obtained in part (c).

Answer to Problem 24PPS

The domain of the function is $\left\{1,2,3,4,5,\mathrm{...}\right\}$ .

The range of the function is $\left\{-3,-1,1,3,5,\mathrm{...}\right\}$

Explanation of Solution

A graph is shown below:

Calculation

The points to graph are represented by $\left(n,f_n\right)$ . So, the points are $\left(1,-3\right),\left(2,-1\right),\left(3,1\right),\left(4,3\right),\left(5,5\right)$ and so on.

The domain of the function is the x -coordinate of the points on the graph

  $\left\{1,2,3,4,5,\mathrm{...}\right\}$ .

The range of the function is the y -coordinate of the points on the graph.

  $\left\{-3,-1,1,3,5,\mathrm{...}\right\}$