Geometric and Arithmetic 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, 4, 6, 8 is an arithmetic sequence with a common difference of 2.

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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 common difference in an arithmetic sequence?

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The common difference in an arithmetic sequence can be found by subtracting any term from the subsequent term. For example, in the sequence 5, 8, 11, the common difference is 8 - 5 = 3.

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

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The common ratio in a geometric sequence can be found by dividing any term by the previous term. For example, in the sequence 2, 6, 18, the common ratio is 6 / 2 = 3.

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What is the nth term formula for an arithmetic sequence?

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The nth term of an arithmetic sequence can be calculated 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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What is the nth term formula for a geometric sequence?

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The nth term of a geometric sequence can be calculated 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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Find the next term in the arithmetic sequence: 7, 10, 13, ...

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The next term is 16, as the common difference is 3.

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Find the next term in the geometric sequence: 5, 15, 45, ...

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The next term is 135, as the common ratio is 3.

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

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The common difference is -3.

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What is the common ratio in the sequence: 81, 27, 9, ...?

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The common ratio is 1/3.