Chapter 12.1, Problem 23E

To determine

To calculate: The next four terms for the arithmetic sequence $5n,-n,-7n,\dots$ .

Answer to Problem 23E

The next four terms for the arithmetic sequence are $-13n,-19n,-25n,-31n$ .

Explanation of Solution

Given information:

The arithmetic sequence $5n,-n,-7n,\dots$ .

Formula used:

A sequence with first term as $a_1$ and next term written as sum of preceding term and common difference then it is called an arithmetic sequence. It is denoted as $a_1,a_1+d,a_1+2d+\dots$ .

Calculation:

Consider the arithmetic sequence $5n,-n,-7n,\dots$ .

Recall that a sequence with first term as $a_1$ and next term written as sum of preceding term and common difference then it is called an arithmetic sequence. It is denoted as $a_1,a_1+d,a_1+2d+\dots$ .

Compute the difference between the consecutive terms,

  $\begin{array}{c}{a}_2-a_1=-n-5n\\ =-6n\end{array}$

And

  $\begin{array}{c}{a}_3-a_2=-7n+n\\ =-6n\end{array}$

Therefore, common difference is $-6n$ .

Now, fourth term is,

  $\begin{array}{c}{a}_4=a_3+\left(-6n\right)\\ =-7n-6n\\ =-13n\end{array}$

Now, fifth term is,

  $\begin{array}{c}{a}_5=a_4+\left(-6n\right)\\ =-13n-6n\\ =-19n\end{array}$

Next, sixth term is,

  $\begin{array}{c}{a}_6=a_5+\left(-6n\right)\\ =-19n-6n\\ =-25n\end{array}$

And, seventh term is,

  $\begin{array}{c}{a}_7=a_6+\left(-6n\right)\\ =-25n-6n\\ =-31n\end{array}$

Thus, the next four terms for the arithmetic sequence are $-13n,-19n,-25n,-31n$ .