Arithmetic & 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. This difference is called the common difference.

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How do you find the explicit formula for an arithmetic sequence?

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The explicit formula for the n^th term of an arithmetic sequence can be expressed as a_n = a₁ + (n-1)d, where a₁ is the first term and d is the common difference.

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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.

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

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The common ratio (r) can be found by dividing any term by the previous term, r = a_n/a_n-1.

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What is the difference between an arithmetic sequence and a geometric sequence?

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In an arithmetic sequence, the difference between consecutive terms is constant, while in a geometric sequence, the ratio between consecutive terms is constant.

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What is the recursive formula for an arithmetic sequence?

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The recursive formula for an arithmetic sequence can be expressed as a_n = a_n-1 + d, where d is the common difference.

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What is the recursive formula for a geometric sequence?

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The recursive formula for a geometric sequence can be expressed as a_n = a_n-1 * r, where r is the common ratio.

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Given the sequence 2, 9, 16, 23, what is the common difference?

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The common difference is 7, calculated as 9 - 2 or 16 - 9.

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What is the explicit formula for the sequence 6, 11, 16, 21?

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The explicit formula is a_n = 6 + 5(n - 1).

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Is the sequence 2, 4, 12, 48 geometric?

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No, this sequence is not geometric because the ratio between consecutive terms is not constant.