Chapter 8.2, Problem 6E

To determine

To verify: the sequence is arithmetic or not by evaluating common difference.

Answer to Problem 6E

The sequence $\frac{1}{2}n+6$ is Arithmetic sequence having common difference of $\frac{1}{2}$.

Explanation of Solution

Given information:

The sequence which needs to be verified is as follows.

  $\frac{1}{2}n+6$

Formula used:

Common difference of the arithmetic sequence is difference of its two consecutive terms.

Calculation:

The sequence is as follows.

  $\frac{1}{2}n+6$

Find the first four terms of the sequence.

for $n=1$ , $a_1=\frac{1}{2}\left(1\right)+6=\frac{1+12}{2}=\frac{13}{2}$

for $n=2$ , $a_2=\frac{1}{2}\left(2\right)+6=1+6=7$

for $n=3$ , $a_3=\frac{1}{2}\left(3\right)+6=\frac{3+12}{2}=\frac{15}{2}$

for $n=4$ , $a_4=\frac{1}{2}\left(4\right)+6=2+6=8$

As $a_2-a_1=7-\frac{13}{2}=\frac{14-13}{2}=\frac{1}{2}$ ,

  $a_3-a_2=\frac{15}{2}-7=\frac{15-14}{2}=\frac{1}{2}$

  $a_4-a_3=8-\frac{15}{2}=\frac{16-15}{2}=\frac{1}{2}$

So common difference $d=a_2-a_1=a_3-a_2=a_4-a_3=\frac{1}{2}$ .

Hence it can be concluded that the sequence $\frac{1}{2}n+6$ is Arithmetic sequence having common difference of $\frac{1}{2}$.