Question

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

3, 1, 13
, 19
, 127
, . . .

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

Answer 1

To classify the sequence \(3, 1, 13, 19, 127, \ldots\), we can analyze the pattern among the numbers provided.

1. **Arithmetic Sequence**: An arithmetic sequence has a constant difference between consecutive terms. For this set:
- \(1 - 3 = -2\)
- \(13 - 1 = 12\)
- \(19 - 13 = 6\)
- \(127 - 19 = 108\)
Since the differences are not constant, this is **not an arithmetic sequence**.

2. **Geometric Sequence**: A geometric sequence has a constant ratio between consecutive terms. For this set:
- \(\frac{1}{3} \approx 0.33\)
- \(\frac{13}{1} = 13\)
- \(\frac{19}{13} \approx 1.46\)
- \(\frac{127}{19} \approx 6.68\)
Since the ratios are not constant, this is **not a geometric sequence**.

3. **Other Types of Sequence**: The sequence may possibly fit another formula or pattern. The terms given do not appear to follow an easily discernible mathematical pattern as either an arithmetic or geometric sequence.

4. **Not a Sequence**: The definition of a sequence is simply an ordered list of numbers. Since there is an ordered list presented, it is indeed a sequence.

Given this analysis, the most appropriate classification for this set of numbers is:

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