Chapter 12.1, Problem 25E

To determine

To calculate: The next four terms for the arithmetic sequence $2a-5+2a+2,2a+9,\dots$ .

Answer to Problem 25E

The next four terms for the arithmetic sequence are $2a+16,2a+23,2a+30,2a+37$ .

Explanation of Solution

Given information:

The arithmetic sequence $2a-5+2a+2,2a+9,\dots$ .

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 $2a-5+2a+2,2a+9,\dots$ .

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,

  $\begin{array}{c}{a}_2-a_1=2a+2-2a+5\\ =7\end{array}$

And

  $\begin{array}{c}{a}_3-a_2=2a+9-2a-2\\ =7\end{array}$

Therefore, common difference is $7$ .

Now, fourth term is,

  $\begin{array}{c}{a}_4=a_3+7\\ =2a+9+7\\ =2a+16\end{array}$

Now, fifth term is,

  $\begin{array}{c}{a}_5=a_4+7\\ =2a+16+7\\ =2a+23\end{array}$

Next, sixth term is,

  $\begin{array}{c}{a}_6=a_5+7\\ =2a+23+7\\ =2a+30\end{array}$

And, seventh term is,

  $\begin{array}{c}{a}_7=a_6+7\\ =2a+30+7\\ =2a+37\end{array}$

Thus, the next four terms for the arithmetic sequence are $2a+16,2a+23,2a+30,2a+37$ .