Chapter 9.2, Problem 36E

To determine

To find :The first five terms of the arithmetic sequence .

Answer to Problem 36E

The first five terms of the arithmetic sequence are

  $a_1=22.45,a_2=20.725,a_3=19,a_4=17.725,a_5=15.55$

Explanation of Solution

Given information: $a_3=19,a_{15}=-1.7$

Calculation:

Let $a_1,d$be first term and common difference of the arithmetic sequence.

We know that the $n$ th term of an arithmetic sequence is

  $a_n=a_1+\left(n-1\right)d$

Therefore, we have

  $\begin{array}{l}{a}_3=a_1+\left(3-1\right)d\\ 19=a_1+2d\mathrm{........}(1)\\ a_{15}=a_1+\left(15-1\right)d\\ -1.7=a_1+14d\mathrm{........}(2)\end{array}$

By subtracting the equation (1) from equation (2), we get

  $\begin{array}{l}-1.7-19=12d\\ -20.7=12d\\ d=-1.725\end{array}$

Also, from equation (1), we have

  $\begin{array}{l}19=a_1+2\times \left(-1.725\right)\\ a_1=22.45\end{array}$

By substituting $n=2,3,4,5$ in the above formula for $n$ th term of the arithmetic sequence and by substituting values of $a_1,d$ , we get

  $\begin{array}{l}{a}_2=a_1+\left(2-1\right)d=22.45+\left(-1.725\right)=20.725\\ a_3=a_1+\left(3-1\right)d=22.45+2\times \left(-1.725\right)=19\\ a_4=a_1+\left(4-1\right)d=22.45+3\times \left(-1.725\right)=17.725\\ a_5=a_1+\left(5-1\right)d=22.45+4\times \left(-1.725\right)=15.55\end{array}$