To analyze the series of numbers Joslyn planted (1, 2, 4, 8, 16, 32), we can look at the characteristics of the sequence and determine which statements are true.
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Is the sequence arithmetic?
- An arithmetic sequence is characterized by a constant difference between consecutive terms. Here, the differences are not constant (1, 2, 4, 8, 16, etc.), so A is false.
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Is the sequence geometric?
- A geometric sequence has a constant ratio between consecutive terms. In this case, each term is doubled (the ratio is 2). Therefore, the sequence is geometric, so B is true.
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Is the sequence represented by \( d = 2^n \)?
- For \( n = 1 \), \( d = 2^1 = 2 \) (2nd term); for \( n = 2 \), \( d = 2^2 = 4 \) (3rd term), and so on. This would not accurately represent the first term (1 is the first term, which corresponds to \( d = 2^0 = 1 \) if we start counting from \( n = 0 \)). Thus C is false.
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Is the sequence represented by \( d = 2^{n-1} \)?
- For \( n = 1 \), \( d = 2^{1-1} = 2^0 = 1 \) (1st term), for \( n = 2 \), \( d = 2^{2-1} = 2^1 = 2 \) (2nd term), etc. This works correctly and matches the provided sequence, so D is true.
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Is the sequence represented by \( d = 2^n \)? (duplicate assessment, needs explicit checking).
- As we previously assessed, this representation does not start correctly at 1 for the first term. Thus, E is false.
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Is the sequence represented by \( d = 2^{n-1} \)? (duplicate assessment, confirmation necessary).
- As we have confirmed working through values previously, and it matches, so this reinforces D is true.
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Is the sequence represented by \( d = 2^n - 1 \)?
- If \( n = 1 \), then \( d = 2^1 - 1 = 1 \) (1st term); however, for other values, this will lead to incorrect results. Hence F is false.
Based on this assessment, the true statements are:
B. The sequence is geometric.
D. The sequence is represented by \( d = 2^{n-1} \).
1 answer