Question

Classify the following set of numbers by determining which of the options is true.

3, 1, 1/3
, 1/9
, 1/27
, . . .

Option #1: The set is an arithmetic sequence.

Option #2: The set is a geometric sequence.

Option #3: The set is some other type of sequence.

Option #4: The set is not a sequence.

(1 point)
The true statement is Option #
.

Answer 1

To classify the set of numbers \(3, 1, \frac{1}{3}, \frac{1}{9}, \frac{1}{27}, \ldots\), we need to determine the relationship between consecutive terms.

1. **Arithmetic Sequence**: In an arithmetic sequence, the difference between consecutive terms is constant.
- Checking the differences:
- \(1 - 3 = -2\)
- \(\frac{1}{3} - 1 = -\frac{2}{3}\)
- \(\frac{1}{9} - \frac{1}{3} = -\frac{2}{9}\)
- \(\frac{1}{27} - \frac{1}{9} = -\frac{2}{27}\)
- The differences are not constant; thus, it is not an arithmetic sequence.

2. **Geometric Sequence**: In a geometric sequence, the ratio of consecutive terms is constant.
- Checking the ratios:
- \(\frac{1}{3} \div 1 = \frac{1}{3}\)
- \(\frac{1}{9} \div \frac{1}{3} = 1/3\)
- \(\frac{1}{27} \div \frac{1}{9} = 1/3\)
- The ratios are constant and equal to \(\frac{1}{3}\); thus, it is a geometric sequence.

Since the sequence meets the condition of having a constant ratio, the correct classification of the sequence is:

**The true statement is Option #2: The set is a geometric sequence.**