Chapter 8.2, Problem 23E

To determine

To calculate: The $nth$ term of the sequence and sketch the graph the first six terms of the sequence.

Answer to Problem 23E

The $nth$ term of the sequence $a_n=5n-12$

The graph of first six points of the sequence is

  

Explanation of Solution

Given information:

  $a_{11}=43,d=5$ .

Formula used:

Arithmetic difference − the difference of consecutive term is constant. The constant difference is called the common difference and is denoted by d.

Therefore,

  $d=a_{n+1}-a_n;n\ge 1$ .

The $nth$ tem of an arithmetic sequence with first term $a_1$ and common difference $d$ is given by : $a_n=a_1+(n-1)d$ .

Calculation:

Step 1. Using general rule to find the first term .

  $\begin{array}{l}=a_n=a_1+(n-1)d\\ =a_{11}=a_1+(11-1)5\\ =43=a_1+\left[\left(11\times 5\right)-\left(1\times 5\right)\right]\\ =43=a_1+55-5\\ =43=a_1+50\\ =43-50=a_1\\ =-7=a_1\end{array}$

Step 2. Rule for $nth$ term.

  $\begin{array}{l}{a}_n=-7+(n-1)5\\ =-7+5n-5\\ =5n-12\end{array}$

Step 3. Using the rule formed in Step 2. will construct a table and then plot the points.

  $a_n=5n-12$

$n$	$1$	$2$	$3$	$4$	$5$	$6$
$a_n$	$\begin{array}{l}{a}_n=5(1)-12\\ =5-12\\ =-7\end{array}$	$\begin{array}{l}{a}_n=5(2)-12\\ =10-12\\ =-2\end{array}$	$\begin{array}{l}{a}_n=5(3)-12\\ =15-12\\ =3\end{array}$	$\begin{array}{l}{a}_n=5(4)-12\\ =20-12\\ =8\end{array}$	$\begin{array}{l}{a}_n=5(5)-12\\ =25-12\\ =13\end{array}$	$\begin{array}{l}{a}_n=5(6)-12\\ =30-12\\ =18\end{array}$

  

Figure 1.

The x-axis is represented as $n$ and y-axis is represented as $a_n$ . The first six points of the sequence in the graph starting from the red dot to orange dot. The sequence is increasing.