Chapter 9, Problem 27RE

To determine

To find :a formula for $a_n$ for the arithmetic sequence..

Answer to Problem 27RE

  $a_n=150-18n$

Explanation of Solution

Given information: $a_3=96,a_7=24$

Calculation:

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

We know that the nth term of the arithmetic sequence is

  $a_n=a_1+\left(n-1\right)d$

Therefore,

  $\begin{array}{l}{a}_3=a_1+\left(3-1\right)d\\ 96=a_1+2d\mathrm{......}\left(1\right)\\ a_7=a_1+\left(7-1\right)d\\ 24=a_1+6d\mathrm{......}\left(2\right)\end{array}$

Subtracting equation (1) from equation (2), we get

  $\begin{array}{l}24-96=4d\\ 4d=-72\\ d=-18\end{array}$

Now, from equation (1)

  $\begin{array}{l}96=a_1+2\left(-18\right)\\ a_1=96+36=132\end{array}$

Thus, the nth term of the arithmetic sequence is

  $\begin{array}{l}{a}_n=132+\left(n-1\right)\left(-18\right)\\ a_n=150-18n\end{array}$