Chapter 9, Problem 28RE

To determine

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

Answer to Problem 28RE

  $a_n=\frac{31}{3}-\frac{1}{3}n$

Explanation of Solution

Given information: $a_7=8,a_{13}=6$

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,

  $\begin{array}{l}{a}_7=a_1+\left(7-1\right)d\\ 8=a_1+6d\mathrm{......}\left(1\right)\\ a_{13}=a_1+\left(13-1\right)d\\ 6=a_1+12d\mathrm{......}\left(2\right)\end{array}$

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

  $\begin{array}{l}6-8=6d\\ 6d=-2\\ d=-\frac{1}{3}\end{array}$

Now, from equation (1)

  $\begin{array}{l}8=a_1+6\left(-\frac{1}{3}\right)\\ a_1=8+2=10\end{array}$

Thus, the nth term of the arithmetic sequence is

  $\begin{array}{l}{a}_n=10+\left(n-1\right)\left(-\frac{1}{3}\right)\\ a_n=\frac{31}{3}-\frac{1}{3}n\end{array}$