Question

forthe following sequence determine the common difference or ratio

18,-6,2

Answer 1

To determine whether the sequence \(18, -6, 2\) has a common difference (indicating it is an arithmetic sequence) or a common ratio (indicating it is a geometric sequence), we can perform the following calculations:

### Arithmetic Sequence:
In an arithmetic sequence, the difference between consecutive terms is constant.

1. Find the difference between the second term and the first term:
\[
-6 - 18 = -24
\]

2. Find the difference between the third term and the second term:
\[
2 - (-6) = 2 + 6 = 8
\]

Since the differences \(-24\) and \(8\) are not equal, this sequence is not arithmetic, and there is no common difference.

### Geometric Sequence:
In a geometric sequence, the ratio between consecutive terms is constant.

1. Find the ratio of the second term to the first term:
\[
\frac{-6}{18} = -\frac{1}{3}
\]

2. Find the ratio of the third term to the second term:
\[
\frac{2}{-6} = -\frac{1}{3}
\]

Since both ratios are equal to \(-\frac{1}{3}\), this sequence is a geometric sequence with a common ratio of \(-\frac{1}{3}\).

### Conclusion:
The sequence \(18, -6, 2\) has a common ratio of \(-\frac{1}{3}\).