Chapter 8.2, Problem 36E

To determine

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

. Verify the result graphically.

Answer to Problem 36E

First five terms of the arithmetic sequence are $-3,-10,-17,-24,-31$ respectively.

Explanation of Solution

Given information:

An sequence is given with sixth term $a_6=-38$ and eleventh term $a_{11}=-73$ .

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 sixth term and the eleventh term.

  $a_6=-38$ and $a_{11}=-73$

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

Now, expression for sixth term will be

  $\begin{array}{c}{a}_6=a_1+\left(6-1\right)d\\ -38=a_1+5d\\ a_1+5d=-38\end{array}$ …… (1)

Now, expression for eleventh term will be

  $\begin{array}{c}{a}_{11}=a_1+\left(11-1\right)d\\ -73=a_1+10d\\ a_1+10d=-73\end{array}$ …… (2)

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

  $\begin{array}{c}{a}_1+10d-a_1-5d=-73+38\\ 5d=35\\ d=7\end{array}$

Now, the first term can be found by substituting $-7$ for d in equation (2).

  $\begin{array}{c}{a}_1+10d=-73\\ a_1+10\left(-7\right)=-73\\ a_1-70=-73\\ a_1=-73+70\\ a_1=-3\end{array}$

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

  $\begin{array}{c}{a}_n=a_1+\left(n-1\right)d\\ =-3+\left(n-1\right)\left(-7\right)\\ =-3-7n+7\\ =4-7n\end{array}$

Thus, the expression for the nth term of the arithmetic sequence is $a_n=4-7n$ .

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

  $\begin{array}{l}{a}_2=a_1+d=-3-7=-10\\ a_3=a_2+d=-10-7=-17\\ a_4=a_3+d=-17-7=-24\\ a_5=a_4+d=-24-7=-31\end{array}$

Hence, the first five terms of the sequence are $-3,-10,-17,-24,-31$ 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.