Chapter 8.2, Problem 37E

To determine

To find: first five terms of the arithmetic sequencefor given third term and 15th term.

. Verify the result graphically.

Answer to Problem 37E

First five terms of the arithmetic sequence are $22.45,20.725,19,17.225,15.55$ respectively.

Explanation of Solution

Given information:

An sequence is given with third term $a_3=19$ and 15th term $a_{15}=-1.7$ .

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 third term and the 15th term.

  $a_3=19$ and $a_{15}=-1.7$

Consider dbe the common difference and $a_1$ as first term.

Now, expression for third term will be

  $\begin{array}{c}{a}_3=a_1+\left(3-1\right)d\\ 19=a_1+2d\\ a_1+2d=19\end{array}$ …… (1)

Now, expression for 15th term will be

  $\begin{array}{c}{a}_{15}=a_1+\left(15-1\right)d\\ -1.7=a_1+14d\\ a_1+14d=-1.7\end{array}$ …… (2)

Subtract equation (1) from equation (2) to find the common difference.

  $\begin{array}{c}{a}_1+14d-a_1-2d=-1.7-19\\ 12d=-20.7\\ d=-1.725\end{array}$

Now, the first term can be found by substituting $-1.725$ for d in equation (2).

  $\begin{array}{c}{a}_1+14d=-1.7\\ a_1+14\left(-1.725\right)=-1.7\\ a_1-24.15=-1.7\\ a_1=-1.7+24.15\\ a_1=22.45\end{array}$

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

  $\begin{array}{c}{a}_n=a_1+\left(n-1\right)d\\ =22.45+\left(n-1\right)\left(-1.725\right)\\ =-22.45-1.725n+1.725\\ =-20.75-1.725n\end{array}$

Thus, the expression for the nth term of the arithmetic sequence is $a_n=-20.75-1.725n$ .

First five terms of the sequence can be found as shown:

  $\begin{array}{l}{a}_2=a_1+d=22.45-1.725=20.725\\ a_3=a_2+d=20.725-1.725=19\\ a_4=a_3+d=19-1.725=17.225\\ a_5=a_4+d=17.225-1.725=15.55\end{array}$

Hence, the first five terms of the sequence are $22.45,20.725,19,17.225,15.55$ respectively.

Use the table feature of graphing utility to find the first 5 terms 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

  

Hence, the result is verified.