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

1 answer

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.