Sequences and Series

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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 (d).

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How do you find the nth term of an arithmetic sequence?

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The nth term (a_n) 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 sum of the first n terms of an arithmetic series?

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

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S_5 = 5/2 * (2 + 12) = 5/2 * 14 = 35.

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What is a recursive sequence?

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A recursive sequence is a sequence in which each term is defined as a function of one or more of the preceding terms.

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How do you find the nth term of a recursive sequence?

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To find the nth term of a recursive sequence, you need the initial term(s) and the recursive formula that relates each term to its predecessors.

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What is the common difference in the sequence 3, 7, 11, 15?

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The common difference (d) is 4, as each term increases by 4 from the previous term.

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Find the 10th term of the arithmetic sequence where a_1=5 and d=3.

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a_{10} = 5 + (10-1) * 3 = 5 + 27 = 32.

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What is the explicit formula for the nth term of the sequence defined by a_1=10 and d=2?

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The explicit formula is a_n = 10 + (n-1) * 2.

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S_4 = 4/2 * (1 + 9) = 2 * 10 = 20.