Chapter 9.2, Problem 29E

To determine

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

Answer to Problem 29E

  $a_n=85+3n$

Explanation of Solution

Given information: $a_3=94,a_6=103$

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,

  $\begin{array}{l}{a}_3=a_1+\left(3-1\right)d\\ 94=a_1+2d\mathrm{........}(1)\\ a_6=a_1+\left(6-1\right)d\\ 103=a_1+5d\mathrm{........}(2)\end{array}$

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

  $\begin{array}{l}103-94=5d-2d\\ 9=3d\\ d=3\end{array}$

Also, from equation (1), we have

  $\begin{array}{l}94=a_1+2\times 3\\ a_1=88\end{array}$

Therefore, formula for the nth term of the arithmetic sequence is

  $\begin{array}{l}{a}_n=a_1+\left(n-1\right)d\\ a_n=88+\left(n-1\right)\left(3\right)\\ a_n=85+3n\end{array}$