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. This difference is called the common difference.

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

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To find the common difference, subtract any term from the subsequent term. For example, in the sequence 25, 13, 1, -11, the common difference is 13 - 25 = -12.

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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 ratio in a geometric sequence?

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To find the common ratio, divide any term by the previous term. For example, in the sequence 5, 10, 20, 40, the common ratio is 10 / 5 = 2.

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What is the formula for 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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What is the formula for 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.

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Find the next three terms in the arithmetic sequence: 3, 7, 11, 15.

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The next three terms are 19, 23, 27.

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What is the 12th term of the sequence: 3, -6, 12, -24, 48?

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The 12th term is -6,144.

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What is the explicit formula for the sequence: 3, 6, 12, 24?

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The explicit formula is a_n = 3 * 2^(n - 1).

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What is the sum of the first n terms of an arithmetic sequence?

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The sum of the first n terms can be calculated using the formula: S_n = n/2 * (a_1 + a_n), where S_n is the sum, a_1 is the first term, and a_n is the nth term.