Question

Determine if the sequence is arithmetic or geometric, whether it has a common difference or common ratio, and what the common difference or common ratio is.

200, 100, 50, 25, 12,5, . . .

Answer 1

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 \(0.5\).