To determine whether the sequence is arithmetic or geometric, we can analyze the relationship between consecutive terms.
- Identify the type of sequence:
- A sequence is arithmetic if the difference between consecutive terms is constant (called the common difference).
- A sequence is geometric if the ratio between consecutive terms is constant (called the common ratio).
Let's look at the given sequence: 256, 64, 16, 4, 1.
- Calculate the ratios between consecutive terms:
- From 256 to 64: \( \frac{64}{256} = \frac{1}{4} \)
- From 64 to 16: \( \frac{16}{64} = \frac{1}{4} \)
- From 16 to 4: \( \frac{4}{16} = \frac{1}{4} \)
- From 4 to 1: \( \frac{1}{4} = \frac{1}{4} \)
Since the ratio between each pair of consecutive terms is constant and equal to \( \frac{1}{4} \), this indicates that the sequence is geometric.
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Common ratio: The common ratio (r) of this geometric sequence is \( \frac{1}{4} \).
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Next three terms: To find the next terms in the sequence, we can continue multiplying the last term by the common ratio:
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The next term after 1: \( 1 \times \frac{1}{4} = \frac{1}{4} \)
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The term after \( \frac{1}{4} \): \( \frac{1}{4} \times \frac{1}{4} = \frac{1}{16} \)
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The term after \( \frac{1}{16} \): \( \frac{1}{16} \times \frac{1}{4} = \frac{1}{64} \)
Thus, the next three terms are \( \frac{1}{4}, \frac{1}{16}, \) and \( \frac{1}{64} \).
Summary:
- The sequence is geometric.
- It has a common ratio.
- The common ratio is \( \frac{1}{4} \).
- The next three terms are \( \frac{1}{4}, \frac{1}{16}, \frac{1}{64} \).