To determine whether the sequence [tex]\(-2, 4, -8, 16\)[/tex] is arithmetic or geometric, we need to examine the pattern of differences (for arithmetic) or ratios (for geometric). Here is the step-by-step analysis:
1. Check if the sequence is arithmetic:
- An arithmetic sequence has a constant difference between consecutive terms.
- Calculate the differences between consecutive terms:
- Difference between 4 and -2: [tex]\(4 - (-2) = 6\)[/tex]
- Difference between -8 and 4: [tex]\(-8 - 4 = -12\)[/tex]
- Difference between 16 and -8: [tex]\(16 - (-8) = 24\)[/tex]
- The differences are [tex]\(6\)[/tex], [tex]\(-12\)[/tex], and [tex]\(24\)[/tex], which are not constant.
Since the differences are not constant, the sequence is not arithmetic.
2. Check if the sequence is geometric:
- A geometric sequence has a constant ratio between consecutive terms.
- Calculate the ratios between consecutive terms:
- Ratio of 4 to -2: [tex]\(4 / (-2) = -2\)[/tex]
- Ratio of -8 to 4: [tex]\(-8 / 4 = -2\)[/tex]
- Ratio of 16 to -8: [tex]\(16 / (-8) = -2\)[/tex]
- The ratios are all [tex]\(-2\)[/tex], which are constant.
Since the ratios are constant, the sequence is geometric.
Therefore, the sequence [tex]\(-2, 4, -8, 16, \ldots\)[/tex] is part of a geometric sequence.
