Chapter 9.2, Problem 18E

To determine

To find the first five terms of the sequence defined by $a_n=n-{\left(-1\right)}^n$ , where $n\ge 1$ and TO-DETERMINE whether the sequence is arithmetic, if yes, then find its common difference.

Answer to Problem 18E

  $a_1=2,a_2=1,a_3=4,a_4=3,a_5=6$

The sequence defined by $a_n=n-{\left(-1\right)}^n$ is not arithmetic.

Explanation of Solution

Given information :

Let the sequence be defined by $a_n=n-{\left(-1\right)}^n$ .

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:

The first five terms of the sequence defined by $a_n=n-{\left(-1\right)}^n$ are:

  $a_1=1-{\left(-1\right)}^1=1+1=2$

  $a_2=2-{\left(-1\right)}^2=1$

  $a_3=3-{\left(-1\right)}^3=3+1=4$

  $\begin{array}{l}{a}_4=4-{\left(-1\right)}^4=3\\ a_5=5-{\left(-1\right)}^5=5+1=6\end{array}$

The terms of the sequence are $a_1=2,a_2=1,a_3=4,a_4=3,a_5=6,\dots ,$ .

In order TO-DETERMINE whether the sequence is arithmetic or not, calculate the difference of the consecutive terms:

  $\begin{array}{l}{a}_2-a_1=1-2=-1\\ a_3-a_2=4-1=3\\ a_4-a_3=3-4=-1\\ a_5-a_4=6-3=3\\ ⋮\end{array}$

And so on.

Since, $a_2-a_1\ne a_3-a_2\ne a_4-a_3\ne a_5-a_4$ , therefore the sequence is not arithmetic.