Question
What kind of sequence is this?
1, 4, 9, 16, ...
1, 4, 9, 16, ...
Answers
Answered by
GPT-4o mini
The sequence you provided is a sequence of perfect squares. Each term in the sequence is the square of a positive integer:
- \(1^2 = 1\)
- \(2^2 = 4\)
- \(3^2 = 9\)
- \(4^2 = 16\)
The general term of the sequence can be expressed as \(n^2\), where \(n\) is a positive integer (1, 2, 3, 4, ...). So the next terms in the sequence would be \(25\) (which is \(5^2\)), \(36\) (which is \(6^2\)), and so on.
- \(1^2 = 1\)
- \(2^2 = 4\)
- \(3^2 = 9\)
- \(4^2 = 16\)
The general term of the sequence can be expressed as \(n^2\), where \(n\) is a positive integer (1, 2, 3, 4, ...). So the next terms in the sequence would be \(25\) (which is \(5^2\)), \(36\) (which is \(6^2\)), and so on.
Answer
but is it arithmetic, geometric, both, neither
Answered by
GPT-4o mini
The sequence \(1, 4, 9, 16, \ldots\) is neither arithmetic nor geometric.
1. **Arithmetic Sequence**: An arithmetic sequence has a constant difference between consecutive terms. In this sequence:
- The difference between the consecutive terms is:
- \(4 - 1 = 3\)
- \(9 - 4 = 5\)
- \(16 - 9 = 7\)
- The differences are not constant (3, 5, 7), so it is not an arithmetic sequence.
2. **Geometric Sequence**: A geometric sequence has a constant ratio between consecutive terms. In this sequence:
- The ratios between consecutive terms are:
- \( \frac{4}{1} = 4\)
- \( \frac{9}{4} = 2.25\)
- \( \frac{16}{9} \approx 1.78\)
- The ratios are not constant, so it is not a geometric sequence.
Since it does not satisfy the conditions for either an arithmetic or geometric sequence, we conclude that the sequence is **neither**. It is a sequence of perfect squares.
1. **Arithmetic Sequence**: An arithmetic sequence has a constant difference between consecutive terms. In this sequence:
- The difference between the consecutive terms is:
- \(4 - 1 = 3\)
- \(9 - 4 = 5\)
- \(16 - 9 = 7\)
- The differences are not constant (3, 5, 7), so it is not an arithmetic sequence.
2. **Geometric Sequence**: A geometric sequence has a constant ratio between consecutive terms. In this sequence:
- The ratios between consecutive terms are:
- \( \frac{4}{1} = 4\)
- \( \frac{9}{4} = 2.25\)
- \( \frac{16}{9} \approx 1.78\)
- The ratios are not constant, so it is not a geometric sequence.
Since it does not satisfy the conditions for either an arithmetic or geometric sequence, we conclude that the sequence is **neither**. It is a sequence of perfect squares.
Answer
Rachel found some empty bins to put cans into for recycling. She put 461 empty cans in the first bin, 461 empty cans in the second bin, 461 empty cans in the third bin, and 461 empty cans in the fourth bin. What kind of sequence is this?