Chapter 8, Problem 34RE

To determine

To find: first five terms of the arithmetic sequencefor given second term and sixth term.

Verify the result graphically.

Answer to Problem 34RE

First five terms of the arithmetic sequence are $12,14,16,18,20$ respectively.

Explanation of Solution

Given information:

An sequence is given with second term $a_2=14$ and sixth term $a_6=22$ .

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 second term and the sixth term.

  $a_2=14$ and $a_6=22$

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

Now, expression for sixth term will be

  $\begin{array}{c}{a}_6=a_1+\left(6-1\right)d\\ 22=a_1+5d\\ a_1+5d=22\end{array}$ …… (1)

Now, expression for second term will be

  $\begin{array}{c}{a}_2=a_1+\left(2-1\right)d\\ 14=a_1+d\\ a_1+d=14\end{array}$ …… (2)

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

  $\begin{array}{c}{a}_1+5d-a_1-d=22-14\\ 4d=8\\ d=2\end{array}$

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

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

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

  $\begin{array}{c}{a}_n=a_1+\left(n-1\right)d\\ =12+\left(n-1\right)\left(2\right)\\ =12+2n-2\\ =10+2n\end{array}$

Thus, the expression for the nth term of the arithmetic sequence is $a_n=10+2n$ .

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

  $\begin{array}{l}{a}_2=a_1+2=12+2=14\\ a_3=a_2+d=14+2=16\\ a_4=a_3+d=16+2=18\\ a_5=a_4+d=18+2=20\end{array}$

Hence, the first five terms of the sequence are $12,14,16,18,20$ 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.