Chapter 12.1, Problem 5CFU

To determine

To explain: The arithmetic sequences among the two $a_n={\left(-1\right)}^n$ and $b_n={\left(-2\right)}^n$ .

Answer to Problem 5CFU

The only arithmetic sequence is $a_n={\left(-1\right)}^n$ . So, student D. is right.

Explanation of Solution

Given information:

The arithmetic sequences among the two $a_n={\left(-1\right)}^n$ and $b_n={\left(-2\right)}^n$ .

Considerarithmetic sequences among the two $a_n={\left(-1\right)}^n$ and $b_n={\left(-2\right)}^n$ .

Substitute the values of n as 1, 2, 3, 4,… to obtain a few terms of the sequences.

For, $a_n={\left(-1\right)}^n$ the terms are $-1,1-1,1,-1,\dots$ .

Compute the difference between the consecutive terms,

  $\begin{array}{c}{a}_2-a_1=1-\left(-1\right)\\ =1+1\\ =2\end{array}$

And

  $\begin{array}{c}{a}_3-a_2=-1-1\\ =-2\end{array}$

The sequence $a_n={\left(-1\right)}^n$ is an oscillatory sequence. The value is $-1$ when n is odd and 1 when n is even.

It has two subsequences in it. When n is odd the difference between each consecutive term is 2 and when n is even the difference between each consecutive term is $-2$ .

Therefore, it can be concluded that $a_n={\left(-1\right)}^n$ is an arithmetic sequence.

For $b_n={\left(-2\right)}^n$ the terms are $-2,4,-8,16,\dots$ .

Compute the difference between the consecutive terms,

  $\begin{array}{c}{a}_2-a_1=4+2\\ =6\end{array}$

And

  $\begin{array}{c}{a}_3-a_2=-8-4\\ =-12\end{array}$

Since, there is such a huge difference between the difference of two consecutive terms. For arithmetic sequence the common difference should be same.

Therefore, it can be concluded that $b_n={\left(-2\right)}^n$ is not an arithmetic sequence.

Thus, the only arithmetic sequence is $a_n={\left(-1\right)}^n$ . So, student D. is right.