Explicit and Recursive Sequences

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What is an explicit sequence?

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An explicit sequence is defined by a formula that allows you to calculate the nth term directly without needing to know the previous terms.

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What is a recursive sequence?

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A recursive sequence is defined by a formula that relates each term to one or more of its preceding terms.

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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.

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What is the common difference in an arithmetic sequence?

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The common difference is the constant amount that each term increases or decreases by in an arithmetic sequence.

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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 calculated using the formula a_n = a_1 + (n-1)d, where d is the common difference.

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Given a recursive formula a_n = a_{n-1} + 5, a_1 = 3, what is a_4?

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a_4 = 3 + 5 + 5 + 5 = 18.

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What is the first term of the sequence defined by a_n = 2n + 1?

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The first term (a_1) is 2(1) + 1 = 3.

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How do you convert a recursive formula to an explicit formula?

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To convert a recursive formula to an explicit formula, identify the pattern in the terms and express the nth term in terms of n.

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What is the 5th term of the sequence defined by a_n = 3n - 2?

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The 5th term (a_5) is 3(5) - 2 = 13.

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

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The sum S_n of the first n terms can be calculated using S_n = n/2 * (a_1 + a_n).