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What kind of sequence is this?Choices:(A) arithmetic(B) geometric(C) both(D) neither
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Q. What kind of sequence is this?Choices:(A) arithmetic(B) geometric(C) both(D) neither
- Verify Differences Uniform: Let's verify if the differences between consecutive terms are uniform. Given sequence: Are the consecutive differences in the sequence equal? , , . The consecutive differences in the sequence are not equal, but they are increasing by each time.
- Check Ratios Equal: Let's check whether the ratios between consecutive terms are equal. Given sequence: Are the ratios between consecutive terms in the sequence equal? is not an integer, is not an integer, is not an integer. The sequence does not have a common ratio.
- Identify Sequence Type: Arithmetic sequence: Consecutive terms have a common difference. Geometric sequence: Consecutive terms have a common ratio. What type of sequence is this? The sequence has increasing differences and no common ratio. However, the differences are increasing by a constant amount, which suggests a pattern. Let's examine the numbers more closely. These numbers are perfect squares: . Each term is the square of consecutive integers, which means the difference between consecutive terms will always be an odd number that increases by each time. This is a special characteristic of perfect squares.
- Determine Correct Choice: Since the sequence is neither arithmetic (no common difference) nor geometric (no common ratio), but it does follow a specific pattern of perfect squares of consecutive integers, the correct choice is neither arithmetic nor geometric.
