Chapter 9.2, Problem 72E

To determine

To identify: The sequence $66,55,44,33,22,\mathrm{...}$ is an arithmetic, a geometric, or neither.

Answer to Problem 72E

The sequence $66,55,44,33,22,\mathrm{...}$ is an arithmetic sequence.

Explanation of Solution

Given information:

The given sequence is $66,55,44,33,22,\mathrm{...}$ .

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 the common ratio for geometric sequence.

Let $a=66,ar=55,a{r}^2=44$ .

  $\begin{array}{l}r=\frac{ar}{a}\\ =\frac{55}{66}\\ =\frac{5}{6}\end{array}$.........(1)

or

  $\begin{array}{l}r=\frac{a{r}^2}{ar}\\ =\frac{44}{55}\\ =\frac{4}{5}\end{array}$.........(2)

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

Check the common difference. Let $a=66,a+d=55,a+2d=44$ .

  $\begin{array}{l}d=\left(a+d\right)-a\\ =55-66\\ =-11\end{array}$.........(3)

or

  $\begin{array}{l}d=\left(a+2d\right)-\left(a+d\right)\\ =44-55\\ =-11\end{array}$.........(4)

From equation (3) and (4), it is clear that the sequence has a common difference of $-11$ .

Therefore, the sequence $66,55,44,33,22,\mathrm{...}$ is an arithmetic sequence.