Chapter 9, Problem 21RE

To determine

To find whether the sequence $5,-1,-7,-13,-19,...$ is arithmetic and if so, then find its common difference.

Answer to Problem 21RE

The sequence $5,-1,-7,-13,-19,...$ is arithmetic and the common difference $d=-6$ .

Explanation of Solution

Given information :

Let the sequence be $5,-1,-7,-13,-19,...$ .

Concept used:

In a sequence $a_1,a_2,a_3,a_4,\dots \dots ,a_n,\dots$ if the difference between the consecutive terms are the same then it is defined as Arithmetic Sequence.

Here, $a_2-a_1=a_3-a_2=a_4-a_3=\dots \dots =d.$ The number $d$ is the common difference of the arithmetic sequence.

Calculations:

Calculate the difference of the consecutive terms:

  $\begin{array}{l}{a}_2-a_1=\left(-1\right)-\left(5\right)=-6\\ a_3-a_2=\left(-7\right)-\left(-1\right)=-6\\ a_4-a_3=\left(-13\right)-\left(-7\right)=-6\\ a_5-a_4=\left(-19\right)-\left(-13\right)=-6\\ ⋮\end{array}$

And so on.

Since, $a_2-a_1=a_3-a_2=a_4-a_3=a_5-a_4=\dots \dots =-6=d$ , therefore the sequence is arithmetic and the common difference $d=-6$ .