Chapter 9.2, Problem 75E

To determine

To identify: The sequence is $\frac{-1}{2},\frac{1}{2},\frac{3}{2},\frac{5}{2},\frac{7}{2},\mathrm{...}$ an arithmetic, a geometric or neither.

Answer to Problem 75E

The sequence $\frac{-1}{2},\frac{1}{2},\frac{3}{2},\frac{5}{2},\frac{7}{2},\mathrm{...}$ is an arithmetic sequence.

Explanation of Solution

Given information:

The given sequence is $\frac{-1}{2},\frac{1}{2},\frac{3}{2},\frac{5}{2},\frac{7}{2},\mathrm{...}$ .

Calculation:

If the sequence has a common difference, it’s an arithmetic sequence. If it has a common ratio, it is a geometric sequence.

Check common ratio for geometric sequence. Let $a=\frac{-1}{2},ar=\frac{1}{2},a{r}^2=\frac{3}{2}$ .

  $\begin{array}{l}r=\frac{ar}{a}\\ =\frac{\left(\frac{1}{2}\right)}{\left(\frac{-1}{2}\right)}\\ =-1\end{array}$.........(1)

or,

  $\begin{array}{l}r=\frac{a{r}^2}{ar}\\ =\frac{\frac{3}{2}}{\frac{1}{2}}\\ =3\end{array}$.........(2)

From equation (1) and (2), it is clear that the sequence has no common ratio. Therefore, it cannot be a geometric sequence.

Check common difference. Let $a=\frac{-1}{2},a+d=\frac{1}{2},a+2d=\frac{3}{2}$ .

  $\begin{array}{c}d=\left(a+d\right)-a\\ =\frac{1}{2}-\left(\frac{-1}{2}\right)\\ =\frac{1}{2}+\frac{1}{2}\\ =\frac{2}{2}\\ =1\end{array}$.........(3)

or,

  $\begin{array}{l}d=\left(a+2d\right)-\left(a+d\right)\\ =\frac{3}{2}-\frac{1}{2}\\ =\frac{2}{2}\\ =1\end{array}$.........(4)

From equation (3) and (4), it is clear that the sequence has a common difference of $1$ .

Therefore, the given sequence $\frac{-1}{2},\frac{1}{2},\frac{3}{2},\frac{5}{2},\frac{7}{2},\mathrm{...}$ is an arithmetic sequence.