Arithmetic and Geometric Sequences

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What is an arithmetic sequence?

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An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. For example, 2, 4, 6, 8 is an arithmetic sequence with a common difference of 2.

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What is a geometric sequence?

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A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. For example, 3, 6, 12, 24 is a geometric sequence with a common ratio of 2.

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How do you find the common difference in an arithmetic sequence?

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The common difference in an arithmetic sequence is found by subtracting any term from the term that follows it. For example, in the sequence 5, 8, 11, the common difference is 8 - 5 = 3.

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How do you find the common ratio in a geometric sequence?

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The common ratio in a geometric sequence is found by dividing any term by the term that precedes it. For example, in the sequence 4, 12, 36, the common ratio is 12 / 4 = 3.

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What is the nth term formula for an arithmetic sequence?

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The nth term of an arithmetic sequence can be found using the formula: a_n = a_1 + (n-1)d, where a_1 is the first term, d is the common difference, and n is the term number.

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What is the nth term formula for a geometric sequence?

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The nth term of a geometric sequence can be found using the formula: a_n = a_1 * r^(n-1), where a_1 is the first term, r is the common ratio, and n is the term number.

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Is the sequence 1, 3, 5, 7, ... arithmetic or geometric?

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The sequence 1, 3, 5, 7 is arithmetic because the difference between consecutive terms is constant (2).

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Is the sequence 2, 4, 8, 16, ... arithmetic or geometric?

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The sequence 2, 4, 8, 16 is geometric because each term is multiplied by 2 to get the next term.

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What happens to the common ratio if a geometric sequence has a term of 0?

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If a geometric sequence has a term of 0, all subsequent terms will also be 0, regardless of the common ratio.

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Can a sequence be both arithmetic and geometric?

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A sequence can be both arithmetic and geometric if all terms are the same (e.g., 5, 5, 5, ...). In this case, the common difference and common ratio are both 0.

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