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, 2, 6, 18, 54 is a geometric sequence with a common ratio of 3.

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

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The common difference (d) in an arithmetic sequence can be found by subtracting any term from the subsequent term. For example, in the sequence 4, 7, 10, the common difference is 7 - 4 = 3.

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

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The common ratio (r) in a geometric sequence can be found by dividing any term by the previous term. For example, in the sequence 3, 9, 27, the common ratio is 9 / 3 = 3.

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What is the nth term formula for 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 nth term formula for 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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Identify whether the sequence 1, 3, 7, 13 is arithmetic, geometric, or neither.

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The sequence 1, 3, 7, 13 is neither arithmetic nor geometric because the differences (2, 4, 6) are not constant and the ratios are not constant.

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What are 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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Find the 16th term of the arithmetic sequence 3, 11, 19, ...

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The 16th term is 25.

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What is the first term of a geometric sequence if the second term is 10 and the common ratio is 2?

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The first term is 5, because 5 * 2 = 10.