To determine whether the sequence \(200, 100, 50, 25, 12.5, \ldots\) is arithmetic or geometric, we need to analyze the differences or ratios between consecutive terms.
1. Checking for an arithmetic sequence: An arithmetic sequence has a common difference, which means the difference between consecutive terms is constant.
Let's calculate the differences:
- \(100 - 200 = -100\)
- \(50 - 100 = -50\)
- \(25 - 50 = -25\)
- \(12.5 - 25 = -12.5\)
The differences are \(-100\), \(-50\), \(-25\), and \(-12.5\), which are not constant. Therefore, the sequence is not arithmetic.
2. Checking for a geometric sequence: A geometric sequence has a common ratio, meaning the ratio between consecutive terms is constant.
Let's calculate the ratios:
- \(\frac{100}{200} = 0.5\)
- \(\frac{50}{100} = 0.5\)
- \(\frac{25}{50} = 0.5\)
- \(\frac{12.5}{25} = 0.5\)
The ratios are all \(0.5\), which are constant. Therefore, the sequence is geometric.
Conclusion: The sequence is geometric with a common ratio of \(0.5\).