is this arithmetic, geometric or nether

2,-6,18,-54,162,-486

1 answer

To determine if the sequence \(2, -6, 18, -54, 162, -486\) is arithmetic, geometric, or neither, we can evaluate the characteristics of each type of sequence.

  1. Arithmetic Sequence: In an arithmetic sequence, the difference between consecutive terms is constant. Let's calculate the differences:

    • \(-6 - 2 = -8\)
    • \(18 - (-6) = 24\)
    • \(-54 - 18 = -72\)
    • \(162 - (-54) = 216\)
    • \(-486 - 162 = -648\)

    The differences are: \(-8, 24, -72, 216, -648\), which are not constant. Therefore, this sequence is not arithmetic.

  2. Geometric Sequence: In a geometric sequence, the ratio between consecutive terms is constant. Let's calculate the ratios:

    • \(\frac{-6}{2} = -3\)
    • \(\frac{18}{-6} = -3\)
    • \(\frac{-54}{18} = -3\)
    • \(\frac{162}{-54} = -3\)
    • \(\frac{-486}{162} = -3\)

    The ratios are all \(-3\), which are constant. Therefore, this sequence is geometric.

Since the sequence has a constant ratio, it is a geometric sequence.