Arithmetic and Geometric Sequences Recursive and Explicit Formulas

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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, in the sequence 3, 6, 9, 12, the common difference is 3.

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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, in the sequence 2, 4, 8, 16, the common ratio is 2.

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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 a_n is the nth term, a_(n-1) is the previous term, and d is the common difference.

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What is the explicit formula for an arithmetic sequence?

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The explicit formula for an arithmetic sequence is given by: a_n = a_1 + (n-1)d, where a_n is the nth term, a_1 is the first term, n is the term number, and 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 a_n is the nth term, a_(n-1) is the previous term, and r is the common ratio.

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What is the explicit formula for a geometric sequence?

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The explicit formula for a geometric sequence is given by: a_n = a_1 * r^(n-1), where a_n is the nth term, a_1 is the first term, n is the term number, and r is the common ratio.

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

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The common difference (d) in an arithmetic sequence can be found by subtracting any term from the subsequent term: d = a_(n) - a_(n-1).

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

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The common ratio (r) in a geometric sequence can be found by dividing any term by the previous term: r = a_(n) / a_(n-1).

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What are the next two terms in the sequence 2, -4, 8, -16?

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The next two terms are 32 and -64.

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What is the common ratio of the sequence 16, -8, 4, -2?

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