Chapter 3.6, Problem 22PPS

a)

To determine

To write: a function to represent the arithmetic sequence.

Answer to Problem 22PPS

A function to represent the arithmetic sequenceis $f\left(n\right)=2n$ .

Explanation of Solution

Given:

The arithmetic sequence is 2, 4, 6,...

Calculation:

Consider the given terms.

2, 4, 6,...

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

  $4-2=6-4=\mathrm{2...}$

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 2 and common difference 2.

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

So, the function for this arithmetic sequence is $f\left(n\right)=2n$ .

b)

To determine

To graph: the function from part (a).

Explanation of Solution

Given:

The arithmetic sequence is 2, 4, 6,...

A function to represent the arithmetic sequence is $f\left(n\right)=2n$ .

Calculation:

The points to graph are represented by $\left(n,f_n\right)$ . So, the points are (1, 2); (2, 4); (3, 6) and so on.

Now, graph the function as shown:

  

c)

To determine

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

Answer to Problem 22PPS

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

The range of the function is $\left\{2,4,6,8,\mathrm{...}\right\}$

Explanation of Solution

Given:

The arithmetic sequence is 2, 4, 6,...

A function to represent the arithmetic sequence is $f\left(n\right)=2n$ .

Calculation:

The points to graph are represented by $\left(n,f_n\right)$ . So, the points are (1, 2); (2, 4); (3, 6) and so on.

The domain of the function is the number of hours spent at the park. So, the domain is $\left\{1,2,3,4,\mathrm{...}\right\}$ .

The range of the function is the number of rides they avail in each interval.

  $\left\{2,4,6,8,\mathrm{...}\right\}$