Chapter 4.6, Problem 57E

(a)

To determine

Whether each sequence is an arithmetic sequence.

Answer to Problem 57E

The sequence $x+6,3x+6,5x+6,7x+6,\mathrm{.......}$ is an arithmetic sequence.

Explanation of Solution

Given:

  $x+6,3x+6,5x+6,7x+6,\mathrm{.......}$

Calculation:

Consider the sequence: $x+6,3x+6,5x+6,7x+6,\mathrm{.......}$

Let us assume $x$ is constant

Now, the difference between the terms is

  $\begin{array}{l}d=a_2-a_1\\ =3x+6-x-6\\ =2x\end{array}$

Similarly,

  $\begin{array}{l}d=a_3-a_2\\ =5x+6-3x-6\\ =2x\end{array}$

And,

  $\begin{array}{l}d=a_4-a_3\\ =7x+6-5x-6\\ =2x\end{array}$

So, the difference between terms is same.

Therefore, the sequence $x+6,3x+6,5x+6,7x+6,\mathrm{.......}$ is an arithmetic sequence.

Conclusion:

Therefore, the sequence $x+6,3x+6,5x+6,7x+6,\mathrm{.......}$ is an arithmetic sequence.

(b)

To determine

Whether each sequence is an arithmetic sequence.

Answer to Problem 57E

The sequence $x+1,3x+1,9x+1,27x+1,\mathrm{........}$ is not an arithmetic sequence.

Explanation of Solution

Given:

  $x+1,3x+1,9x+1,27x+1,\mathrm{........}$

Calculation:

Consider the sequence: $x+1,3x+1,9x+1,27x+1,\mathrm{........}$

Let us assume constant is $1$ , so the sequence will be $2,4,10,28,\mathrm{.......}$

Now, the difference between the terms is

  $\begin{array}{l}d=a_2-a_1\\ =3x+1-x-1\\ =2x\end{array}$

Similarly,

  $\begin{array}{l}d=a_3-a_2\\ =9x+1-3x-1\\ =6x\end{array}$

And,

  $\begin{array}{l}d=a_4-a_3\\ =27x+1-9x-1\\ =18x\end{array}$

So, the difference between terms is not same.

Therefore, the sequence $x+1,3x+1,9x+1,27x+1,\mathrm{........}$ is not an arithmetic sequence.

Conclusion:

Therefore, the sequence $x+1,3x+1,9x+1,27x+1,\mathrm{........}$ is not an arithmetic sequence.