Chapter 12.1, Problem 24E

To determine

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

Answer to Problem 24E

The next four terms for the arithmetic sequence are $5-2k,5-3k,5-4k,5-5k$ .

Explanation of Solution

Given information:

The arithmetic sequence $5+k,5,5-k,\dots$ .

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 $5+k,5,5-k,\dots$ .

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,

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

And

  $\begin{array}{c}{a}_3-a_2=5-k-5\\ =-k\end{array}$

Therefore, common difference is $-k$ .

Now, fourth term is,

  $\begin{array}{c}{a}_4=a_3+\left(-k\right)\\ =5-k-k\\ =5-2k\end{array}$

Now, fifth term is,

  $\begin{array}{c}{a}_5=a_4+\left(-k\right)\\ =5-2k-k\\ =5-3k\end{array}$

Next, sixth term is,

  $\begin{array}{c}{a}_6=a_5+\left(-k\right)\\ =5-3k-k\\ =5-4k\end{array}$

And, seventh term is,

  $\begin{array}{c}{a}_7=a_6+\left(-k\right)\\ =5-4k-k\\ =5-5k\end{array}$

Thus, the next four terms for the arithmetic sequence are $5-2k,5-3k,5-4k,5-5k$ .