Chapter 3.6, Problem 23PPS

a.

To determine

To determine the function of the arithmetic sequence.

Answer to Problem 23PPS

  $a_n=8n$

Explanation of Solution

Given:

Sequence:

  $8,16,24,32,40,48,\mathrm{56...}$

Formula used:

nth term of any arithmetic progression is given by the formula:

  $t_n=a+(n-1)d$

  $t_n=$ nth term

  $a=$ The first term of the arithmetic progression

  $d=$ The common difference of the arithmetic progression

Calculation:

The common difference of the arithmetic progression is:

  $16-8=8$

Applying above formula,

  $\Rightarrow a_n=a+(n-1)d$

  $\Rightarrow a_n=8+(n-1)(8)$  $[\because a=8\&d=8]$

  $\begin{array}{l}\Rightarrow a_n=8+8n-8\\ \Rightarrow a_n=8n\end{array}$

Conclusion:

The required function of the sequence is $a_n=8n$ .

b.

To determine

To graph the function.

Answer to Problem 23PPS

  $a_n=8n$

Explanation of Solution

Given:

Sequence:

  $8,16,24,32,40,48,\mathrm{56..}$

Calculation for graph:

  $(n,a_n)$ represent the points of the graph.

First seven points are: $(1,8),(2,16),(3,24),(4,32),(5,40),(6,48),(7,56)$

Graph:

  

Interpretation:

From the graph it can be observed that the horizontal extent of the graph is from $-\infty$ to $\infty$ .

And the vertical extent of the graph is also from $-\infty$ to $\infty$ .

So, both the domain and range of the function is $(-\infty ,\infty )$ .

c.

To determine

To determine the domain and range of the function

Answer to Problem 23PPS

  $D=\{1,2,3,\mathrm{4...}\}\&R=\{8,16,24,32,\mathrm{...}\}$

Explanation of Solution

Given:

Sequence:

  $8,16,24,32,40,48,\mathrm{56...}$

Calculation:

The domain of the function is the number of ten foot $2\times 4$ plank cut.

The range of the function is the amount paid in the commission.

So, the domain and range are respectively: $D=\{1,2,3,\mathrm{4...}\}\&R=\{8,16,24,32,\mathrm{...}\}.$

Conclusion:

Therefore, the domain and range are respectively: $D=\{1,2,3,\mathrm{4...}\}\&R=\{8,16,24,32,\mathrm{...}\}.$