Question

which of the following is a geometric sequence with a common ratio of \\(\frac{1}{3}\\)? 5 points

a. \\(18, 6, -6, -18, \dots\\)
b. \\(9, 8\frac{2}{3}, 8\frac{1}{3}, 8, \dots\\)
c. \\(1, 3, 9, 27, \dots\\)
d. \\(81, 27, 9, 3, \dots\\)

Explanation:

Define geometric sequence

A geometric sequence is a sequence where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio \(r\).

State common ratio formula

To find the common ratio \(r\), we divide any term by its preceding term:

$$r = \frac{a_{n}}{a_{n-1}}$$

Analyze option A

The sequence is \(18, 6, -6, -18, \dots\).
Calculate successive ratios:

$$\frac{6}{18} = \frac{1}{3}$$

$$\frac{-6}{6} = -1$$

Since the ratios are not equal, this is not a geometric sequence.

Analyze option B

The sequence is \(9, 8\frac{2}{3}, 8\frac{1}{3}, 8, \dots\).
Calculate successive differences:

$$8\frac{2}{3} - 9 = -\frac{1}{3}$$

$$8\frac{1}{3} - 8\frac{2}{3} = -\frac{1}{3}$$

This is an arithmetic sequence with a common difference of \(-\frac{1}{3}\), not a geometric sequence.

Analyze options C and D

For option C: \(1, 3, 9, 27, \dots\)

$$r = \frac{3}{1} = 3$$

The common ratio is \(3\).

For option D: \(81, 27, 9, 3, \dots\)

$$r = \frac{27}{81} = \frac{1}{3}$$

$$r = \frac{9}{27} = \frac{1}{3}$$

$$r = \frac{3}{9} = \frac{1}{3}$$

The common ratio is indeed \(\frac{1}{3}\).

Answer:

• (A) \(18, 6, -6, -18, \dots\)
• (B) \(9, 8\frac{2}{3}, 8\frac{1}{3}, 8, \dots\)
• (C) \(1, 3, 9, 27, \dots\)
• (D) \(81, 27, 9, 3, \dots\) (Correct answer)