Chapter 3.5, Problem 22PPS

(a)

To determine

Write a function to represent the arithmetic sequence.

Answer to Problem 22PPS

The function for this arithmetic sequence is $f_n=2n$

Explanation of Solution

Given: The arithmetic sequence is $2,4,6,\mathrm{......}$

Concept Used:

The arithmetic sequence is $2,4,6,\mathrm{......}$

First find the common difference and then find the nth terms of the sequence.

Formula for nth term is the function of the sequence.

Calculation:

The arithmetic sequence is $2,4,6,\mathrm{......}$

  $a_1=2a_2=4a_3=6$

Find the common difference: $d=a_2-a_1=4-2=2\text{or}d=a_3-a_2=6-4=2$

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.

  $a_1=2;\text{and}d=2$

  $\begin{array}{l}{a}_n=a_1+(n-1)d\text{Formula}\text{for}\text{the}nth\text{term}\\ a_n=2+(n-1)·2a_1=2andd=2\\ a_n=2+2n-2\text{Distributive}\text{Property}\\ a_n=2n\text{Simplify}.\end{array}$

Formula for nth term is always defined as the function.

The function for this arithmetic sequence is $f_n=2n$

Thus, the function for this arithmetic sequence is $f_n=2n$

(b)

To determine

Graph the function and determine the domain.

Answer to Problem 22PPS

The domain is {1, 2, 3, 4, ......}

Explanation of Solution

Given: The arithmetic sequence is $2,4,6,\mathrm{......}$

Concept Used:

The points on the graph are represented by $(n,f_n)$ . So, the points are (1, 2), (2, 4), (3, 6).

Calculation:

The points to graph are represented by $(n,f_n)$ . So, the points are (1, 2), (2, 4), (3, 6).

  

The domain of the function is the number of hours spent at the park.

So, the domain is {1, 2, 3, 4, ......}

Thus, the domain is {1, 2, 3, 4, ......}