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, 5, 8, 11 is an arithmetic sequence with a common difference of 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, 3, 6, 12, 24 is a geometric sequence with a common ratio of 2.

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How do you find the nth term of 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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How do you find the nth term of 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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What is the common difference in an arithmetic sequence?

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The common difference in an arithmetic sequence is the constant amount that each term increases or decreases by. For example, in the sequence 4, 7, 10, the common difference is 3.

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What is the common ratio in a geometric sequence?

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The common ratio in a geometric sequence is the factor by which each term is multiplied to get the next term. For example, in the sequence 2, 6, 18, the common ratio is 3.

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Identify if the sequence 5, 10, 15, 20 is arithmetic or geometric.

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The sequence 5, 10, 15, 20 is an arithmetic sequence because the difference between consecutive terms is constant (5).

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Identify if the sequence 2, 4, 8, 16 is arithmetic or geometric.

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

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What are the next three terms in the arithmetic sequence: 1, 4, 7, ...?

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The next three terms are 10, 13, 16.

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What are the next three terms in the geometric sequence: 3, 9, 27, ...?

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The next three terms are 81, 243, 729.