Chapter 8.2, Problem 35E

To determine

To calculate: The rule for $nth$ term of the arithmetic sequence.

Answer to Problem 35E

The $nth$ term is $a_n=-4n+13$ .

Explanation of Solution

Given information:

  $a_{18}=-59,a_{21}=-71$ .

Formula used:

Arithmetic difference − the difference of consecutive term is constant. The constant difference is called the common difference and is denoted by d.

Therefore,

  $d=a_{n+1}-a_n;n\ge 1$ .

The $nth$ tem of an arithmetic sequence with first term $a_1$ and common difference $d$ is given by : $a_n=a_1+(n-1)d$ .

Calculation:

Consider,

  $a_{18}=-59,a_{21}=-71$

Step 1. System of equations using formula $a_n=a_1+(n-1)d$ .

  $\begin{array}{l}=a_{18}=a_1+(18-1)d\\ =a_{18}=a_1+17d\\ =-59=a_1+17d\text{ (i) }\end{array}$

  $\begin{array}{l}=a_{21}=a_1+(21-1)d\\ =a_{21}=a_1+20d\\ =-71=a_1+20d\text{ (ii) }\end{array}$

Step 2. Solving equation (i) and (ii) we get,

  $\begin{array}{l}=\begin{array}{cccc}-59& =& a_1+& 17d\end{array}\\ \begin{array}{ccccc}=& -71& =& a_1+& 20d\end{array}\to \text{Subtract}\\ =\begin{array}{cccc}\text{ 12}& =& 0& -\text{ 3}d\end{array}\end{array}$

  $\begin{array}{l}=12=-3d\\ =\frac{12}{-3}=\frac{-3d}{-3}\\ =-4=d\end{array}$

Putting value d in one the equation we get,

  $\begin{array}{l}=-59=a_1+17(-4)\\ =-59=a_1-68\\ =-59+68=a_1\\ =9=a_1\end{array}$

Step 3. The general rule.

  $\begin{array}{l}{a}_n=9+(n-1)(-4)\\ =9-4n+4\\ =-4n+13\end{array}$

Hence, the $nth$ term is $a_n=-4n+13$ .