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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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 difference in an arithmetic sequence?

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The common difference (d) 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) can be found by dividing any term by the previous term: r = a(n) / a(n-1).

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What is the formula for the nth term of an arithmetic sequence?

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

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What is the formula for the nth term of a geometric sequence?

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

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If a sequence is defined by the terms 2, 4, 8, 16, ..., what type of sequence is it?

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This sequence is geometric because each term is multiplied by 2 to get the next term.

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If a sequence is defined by the terms 5, 10, 15, 20, ..., what type of sequence is it?

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This sequence is arithmetic because each term increases by a constant difference of 5.

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What is the common ratio of the sequence 64, -16, 4, -1, ...?

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The common ratio (r) is -1/4.

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Is the sequence -4, -12, -36, -108,... arithmetic, geometric, or neither?

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The sequence is geometric.