Question
determine whether the sequence is arithmetic, geometric, or neither.
1.5, 4.5, 13.5, 40.5, ...
1.5, 4.5, 13.5, 40.5, ...
Answers
Answered by
GPT-4o mini
To determine whether the sequence \(1.5, 4.5, 13.5, 40.5, \ldots\) is arithmetic, geometric, or neither, we can examine the pattern of the terms.
1. **Arithmetic Sequence**: An arithmetic sequence has a common difference. We can find the differences between consecutive terms:
- \(4.5 - 1.5 = 3\)
- \(13.5 - 4.5 = 9\)
- \(40.5 - 13.5 = 27\)
The differences are \(3, 9, 27\). Since these differences are not constant, the sequence is not arithmetic.
2. **Geometric Sequence**: A geometric sequence has a common ratio. We can find the ratios between consecutive terms:
- \(\frac{4.5}{1.5} = 3\)
- \(\frac{13.5}{4.5} = 3\)
- \(\frac{40.5}{13.5} = 3\)
The ratios are all \(3\), which is constant. This means the sequence is geometric with a common ratio of \(3\).
Since the sequence exhibits a constant ratio, we conclude that the sequence \(1.5, 4.5, 13.5, 40.5, \ldots\) is a **geometric sequence** with a common ratio of \(3\).
1. **Arithmetic Sequence**: An arithmetic sequence has a common difference. We can find the differences between consecutive terms:
- \(4.5 - 1.5 = 3\)
- \(13.5 - 4.5 = 9\)
- \(40.5 - 13.5 = 27\)
The differences are \(3, 9, 27\). Since these differences are not constant, the sequence is not arithmetic.
2. **Geometric Sequence**: A geometric sequence has a common ratio. We can find the ratios between consecutive terms:
- \(\frac{4.5}{1.5} = 3\)
- \(\frac{13.5}{4.5} = 3\)
- \(\frac{40.5}{13.5} = 3\)
The ratios are all \(3\), which is constant. This means the sequence is geometric with a common ratio of \(3\).
Since the sequence exhibits a constant ratio, we conclude that the sequence \(1.5, 4.5, 13.5, 40.5, \ldots\) is a **geometric sequence** with a common ratio of \(3\).