Chapter 8.2, Problem 45E

To determine

To find: the 7^th termof the arithmetic sequence for given first two terms.

. Verify the result graphically.

Answer to Problem 45E

The 7^thterm of the arithmetic sequence is $-10.2$ .

Explanation of Solution

Given information:

An sequence is given with first term $a_1=4.2$ and second term $a_2=1.8$ .

Concept used:

An arithmetic sequence of n terms, has the form $a_1,a_2,a_3,\mathrm{...},a_n$ and should have a common difference between each two consecutive terms.

That is

  $a_2-a_1=a_3-a_2=a_4-a_3=\mathrm{...}=a_n-a_{n-1}$

Common difference can be defined by d .

  $a_2-a_1=a_3-a_2=a_4-a_3=\mathrm{...}=a_n-a_{n-1}=d$

nth term of the arithmetic sequence has the form

  $a_n=a_1+\left(n-1\right)d$

Where $a_1$ is the first term of the sequence, and d is the common difference.

Calculation:

Now, consider the first and second term of the sequence.

  $a_1=4.2$ and $a_2=1.8$

Consider dbe the common difference.

The common difference can be found by subtracting first term from second term.

  $\begin{array}{c}d=a_2-a_1\\ =1.8-4.2\\ =-2.4\end{array}$

Now, the 7^th term will be

  $\begin{array}{c}{a}_n=a_1+\left(n-1\right)d\\ a_7=4.2+\left(7-1\right)\left(-2.4\right)\\ =4.2-16.8+2.4\\ =6.6-16.8\\ =-10.2\end{array}$

So, the nth term of the arithmetic sequence can be found as

  $\begin{array}{c}{a}_n=a_1+\left(n-1\right)d\\ =4.2+\left(n-1\right)\left(-2.4\right)\\ =4.2-2.4n+2.4\\ =6.6-2.4n\end{array}$

Thus, the expression for the nth term of the arithmetic sequence is $a_n=6.6-2.4n$ .

Use the table feature of graphing utility to find the 7^th term of sequence by using following steps:

Press MODE key, select Seq in the fourth line and hit Enter

  

Press $Y=$ and Enter the sequence,

  

Set the table by pressing TBLSET and then press TABLE,

$n$	$a_n$
1	$4.2$
2	$1.8$
3	$-0.6$
4	$-3$
5	$-5.4$
6	$-7.8$
7	$-10.2$
8	$-12.6$
9	$-15$
10	$-17.4$

Hence, the result is verified.