Chapter 7.2, Problem 52E

Solution

To identify: The similarities and differences between the graphs of a linear function and an arithmetic sequence. The graphs of $a_n=3n+2\text{and }f\left(x\right)=3x+2$ where $n$ is a positive integer and $x$ is a real number.

Similarity: The sequence's terms are the points found on the graph of the specified function.

Difference: Function can take all real values while the sequence has only positive values.

Given information:

The given sequence and function are:

  $a_n=3n+2\text{ and }f\left(x\right)=3x+2$

Calculation:

Consider the sequence $a_n=3n+2$ .

Put the positive integers into $a_n=3n+2$ .

  $\begin{array}{l}{a}_1=3\times 1+2\\ a_1=5\\ a_2=3\times 2+2\\ a_2=8\\ a_3=3\times 3+2\\ a_3=11\\ a_4=3\times 4+2\\ a_4=14\\ a_5=3\times 5+2\\ a_5=17\end{array}$

Draw the graph from given function $f\left(x\right)=3x+2$

  

Put the point of $a_n=3n+2$ into this graph.

  

It can be observed that the sequence's terms are the points found on the graph of the specified function.

Difference: Function can take all real values while the sequence has only positive values.