Chapter 8.2, Problem 19E

To determine

To write: the first five terms of the given sequence. Whether the given sequence is an arithmetic sequence. The common difference for an arithmetic sequence.

Answer to Problem 19E

The first five terms of the given sequence are $\frac{1}{4},\frac{1}{4},\frac{1}{4},\frac{1}{4},\frac{1}{4}$ respectively.

The given sequence is an arithmetic sequence.

The common difference of the arithmetic sequence is $0$ .

Explanation of Solution

Given information:

An sequence as $a_n=\frac{{\left(-1\right)}^{2n}}{4}$ .

Concept used:

An arithmetic sequence of n terms, has the form $a_1,a_2,a_3,\mathrm{...},a_n$ and should have a common difference between each two consecutive terms.

That is

  $a_2-a_1=a_3-a_2=a_4-a_3=\mathrm{...}=a_n-a_{n-1}$

Common difference can be defined by d .

  $a_2-a_1=a_3-a_2=a_4-a_3=\mathrm{...}=a_n-a_{n-1}=d$

nth term of the arithmetic sequence has the form

  $a_n=a_1+\left(n-1\right)d$

Where $a_1$ is the first term of the sequence, and d is the common difference.

Calculation:

Now, consider the given sequence.

  $a_n=\frac{{\left(-1\right)}^{2n}}{4}$

First five terms of the sequence can be found by substituting 1, 2, 3, 4, and 5 for nas shown:

  $\begin{array}{l}{a}_1=\frac{{\left(-1\right)}^2}{4}=\frac{1}{4}\\ a_2=\frac{{\left(-1\right)}^4}{4}=\frac{1}{4}\\ a_3=\frac{{\left(-1\right)}^6}{4}=\frac{1}{4}\\ a_4=\frac{{\left(-1\right)}^8}{4}=\frac{1}{4}\\ a_5=\frac{{\left(-1\right)}^{10}}{4}=\frac{1}{4}\end{array}$

Hence, the first five terms of the sequence are $\frac{1}{4},\frac{1}{4},\frac{1}{4},\frac{1}{4},\frac{1}{4}$ respectively.

Now, the difference can be calculated as shown:

  $\begin{array}{l}{d}_1=a_2-a_1=\frac{1}{4}-\frac{1}{4}=0\\ d_2=a_3-a_2=\frac{1}{4}-\frac{1}{4}=0\\ d_3=a_4-a_3=\frac{1}{4}-\frac{1}{4}=0\\ d_4=a_5-a_4=\frac{1}{4}-\frac{1}{4}=0\end{array}$

Now, the differences for each two consecutive terms are as

  $d_1=0=d_2=d_3=\mathrm{...}$

Therefore, differences are common for each two consecutive terms of the sequence.

So, it is an arithmetic sequence.

And, the common difference of the arithmetic sequence is $d=0$ .