Chapter 8.2, Problem 28E

To determine

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

Answer to Problem 28E

The $nth$ term of the sequence $a_n=-\frac{3}{2}n+\frac{13}{2}$

The graph of first six points of the sequence is

  

Explanation of Solution

Given information:

  $a_{21}=-25,d=-\frac{3}{2}$ .

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 .

  $\begin{array}{l}=a_n=a_1+(n-1)d\\ =a_{21}=a_1+\left(21-1\right)\left(-\frac{3}{2}\right)\\ =-25=a_1+\left[\left(21\times -\frac{3}{2}\right)-\left(1\times -\frac{3}{2}\right)\right]\\ =-25=a_1+\left(-\frac{63}{2}+\frac{3}{2}\right)\\ =-25=a_1+\left(\frac{-60}{2}\right)\\ =-25=a_1-30\\ =-25+30=a_1\\ =5=a_1\end{array}$

Step 2. Rule for $nth$ term.

  $\begin{array}{l}{a}_n=5+\left(n-1\right)\left(-\frac{3}{2}\right)\\ =5-\frac{3}{2}n+\frac{3}{2}\\ =-\frac{3}{2}n+\frac{10+3}{2}\\ =-\frac{3}{2}n+\frac{13}{2}\end{array}$

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

  $a_n=-\frac{3}{2}n+\frac{13}{2}$

$n$	$1$	$2$	$3$	$4$	$5$	$6$
$a_n$	$\begin{array}{l}{a}_n=-\frac{3}{2}(1)+\frac{13}{2}\\ =-\frac{3}{2}+\frac{13}{2}\\ =\frac{10}{2}\\ =5\end{array}$	$\begin{array}{l}{a}_n=-\frac{3}{2}(2)+\frac{13}{2}\\ =-\frac{6}{2}+\frac{13}{2}\\ =\frac{7}{2}\\ =3.5\end{array}$	$\begin{array}{l}{a}_n=-\frac{3}{2}(3)+\frac{13}{2}\\ =-\frac{9}{2}+\frac{13}{2}\\ =\frac{4}{2}\\ =2\end{array}$	$\begin{array}{l}{a}_n=-\frac{3}{2}(4)+\frac{13}{2}\\ =-\frac{12}{2}+\frac{13}{2}\\ =\frac{1}{2}\\ =0.5\end{array}$	$\begin{array}{l}{a}_n=-\frac{3}{2}(5)+\frac{13}{2}\\ =-\frac{15}{2}+\frac{13}{2}\\ =-\frac{2}{2}\\ =-1\end{array}$	$\begin{array}{l}{a}_n=-\frac{3}{2}(6)+\frac{13}{2}\\ =-\frac{18}{2}+\frac{13}{2}\\ =-\frac{5}{2}\\ =-2.5\end{array}$

  

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 decreasing.