Chapter 8.2, Problem 35E

To determine

To find: first five terms of the arithmetic sequence for given eighth term and twelfth term.

. Verify the result graphically.

Answer to Problem 35E

First five terms of the arithmetic sequence are $20,24,28,32,36$ respectively.

Explanation of Solution

Given information:

An sequence is given with eighth term $a_6=26$ and twelfth term $a_{12}=42$ .

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 eighth term and the twelfth term.

  $a_8=26$ and $a_{12}=42$

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

Now, expression for eighth term will be

  $\begin{array}{c}{a}_8=a_1+\left(8-1\right)d\\ 26=a_1+7d\\ a_1+7d=26\end{array}$ …… (1)

Now, expression for twelfth term will be

  $\begin{array}{c}{a}_{12}=a_1+\left(12-1\right)d\\ 42=a_1+11d\\ a_1+11d=42\end{array}$ …… (2)

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

  $\begin{array}{c}{a}_1+11d-a_1-7d=42-26\\ 4d=16\\ d=4\end{array}$

Now, the first term can be found by substituting $4$ for d in equation (2).

  $\begin{array}{c}{a}_1+11d=42\\ a_1+11\left(2\right)=42\\ a_1+22=42\\ a_1=42-22\\ a_1=20\end{array}$

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

  $\begin{array}{c}{a}_n=a_1+\left(n-1\right)d\\ =20+\left(n-1\right)\left(4\right)\\ =20+4n-4\\ =16+4n\end{array}$

Thus, the expression for the nth term of the arithmetic sequence is $a_n=16+4n$ .

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

  $\begin{array}{l}{a}_2=a_1+d=20+4=24\\ a_3=a_2+d=24+4=28\\ a_4=a_3+d=28+4=32\\ a_5=a_4+d=32+4=36\end{array}$

Hence, the first five terms of the sequence are $20,24,28,32,36$ 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.