Question

is this arithmetic, geometric or nether
2,-6,18,-54,162,-486

Answer 1

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**.