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Answer:
An arithmetic sequence has a constant difference between consecutive terms. Let's check for a common difference:
[tex]8 - 1 = 7[/tex]
[tex]27 - 8 = 19[/tex]
Since the differences are not equal, (1, 8, 27) is not an arithmetic sequence.
A geometric sequence has a constant ratio between consecutive terms. Let's check for a common ratio:
[tex]\frac{8}{1} = 8[/tex]
[tex]\frac{27}{8} \approx 3.375[/tex]
Since the ratios are not equal, (1, 8, 27) is not a geometric sequence.
Other Type: Polynomial Sequence
This sequence can be identified as a sequence of cubes. Let's analyze the terms:
[tex]1 = 1^3[/tex]
[tex]8 = 2^3[/tex]
[tex]27 = 3^3[/tex]
The given sequence (1, 8, 27) is the sequence of the cubes of the first three positive integers:
[tex]n^3 \text{ where } n = 1, 2, 3, \ldots[/tex]
Therefore, the next term in this sequence (for ( n = 4 )) would be:
[tex]4^3 = 64[/tex]
- Arithmetic Sequence: No, because it does not have a constant difference.
- Geometric Sequence: No, because it does not have a constant ratio.
- Polynomial Sequence (Cubic): Yes, the sequence represents the cubes of natural numbers ( n³ ).
Thus, the sequence (1, 8, 27) is best described as a sequence of cubes.