Arithmetic and Geometric Sequences

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What is an arithmetic sequence?

Back

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?

Back

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 and d is the common difference.

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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 and r is the common ratio.

Front

What is the common difference in an arithmetic sequence?

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The common difference is the fixed amount that each term increases or decreases by in an arithmetic sequence. 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 is the fixed factor by which each term is multiplied to get the next term in a geometric sequence. For example, in the sequence 2, 6, 18, the common ratio is 3.

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Find the 5th term of the arithmetic sequence: 10, 15, 20, ...

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a_5 = 10 + (5-1)5 = 10 + 20 = 30.

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Find the 4th term of the geometric sequence: 5, 15, 45, ...

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a_4 = 5 * 3^(4-1) = 5 * 27 = 135.

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What is the explicit formula for the arithmetic sequence: 2, 5, 8, 11, ...?

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a_n = 2 + (n-1)3.

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What is the explicit formula for the geometric sequence: 1, 3, 9, 27, ...?

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a_n = 1 * 3^(n-1).