Chapter 8.2, Problem 38E

To determine

To find: first five terms of the arithmetic sequencefor given fifth term and 14th term.

. Verify the result graphically.

Answer to Problem 38E

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 fifth term $a_5=16$ and 14thterm $a_{14}=38.5$ .

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 fifth term and the 14th term.

  $a_5=16$ and $a_{14}=38.5$

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

Now, expression for fifth term will be

  $\begin{array}{c}{a}_5=a_1+\left(5-1\right)d\\ 16=a_1+4d\\ a_1+4d=16\end{array}$ …… (1)

Now, expression for 14th term will be

  $\begin{array}{c}{a}_{15}=a_1+\left(14-1\right)d\\ 38.5=a_1+13d\\ a_1+13d=38.5\end{array}$ …… (2)

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

  $\begin{array}{c}{a}_1+13d-a_1-4d=38.5-16\\ 9d=22.5\\ d=2.5\end{array}$

Now, the first term can be found by substituting $2.5$ for d in equation (2).

  $\begin{array}{c}{a}_1+13d=38.5\\ a_1+13\left(2.5\right)=38.5\\ a_1+32.5=38.5\\ a_1=38.5-32.5\\ a_1=6\end{array}$

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

  $\begin{array}{c}{a}_n=a_1+\left(n-1\right)d\\ =6+\left(n-1\right)\left(2.5\right)\\ =6+2.5n-2.5\\ =3.5+2.5n\end{array}$

Thus, the expression for the nth term of the arithmetic sequence is $a_n=3.5+2.5n$ .

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

  $\begin{array}{l}{a}_2=a_1+d=6+2.5=8.5\\ a_3=a_2+d=8.5+2.5=11\\ a_4=a_3+d=11+2.5=13.5\\ a_5=a_4+d=13.5+2.5=16\end{array}$

Hence, the first five terms of the sequence are $6,8.5,11,13.5,16$ 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.