Explore Arithmetic 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.

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What is the formula for 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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How do you find the common difference in an arithmetic sequence?

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The common difference (d) can be found by subtracting any term from the subsequent term: d = a_(n+1) - a_n.

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What is an arithmetic series?

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An arithmetic series is the sum of the terms of an arithmetic sequence.

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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_1 is the first term and a_n is the nth term.

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If the first term of an arithmetic sequence is 3 and the common difference is 5, what is the 10th term?

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The 10th term can be calculated as: a_10 = 3 + (10 - 1) * 5 = 3 + 45 = 48.

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What is the sum of the first 5 terms of the arithmetic sequence 2, 5, 8, ...?

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The sum of the first 5 terms is: S_5 = 5/2 * (2 + 14) = 5/2 * 16 = 40.

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What is the common difference in the sequence 10, 7, 4, 1, ...?

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The common difference is d = 7 - 10 = -3.

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How can you determine if a sequence is arithmetic?

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A sequence is arithmetic if the difference between consecutive terms is constant.

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What is the 5th term of the arithmetic sequence where a_1 = 4 and d = 2?

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The 5th term is: a_5 = 4 + (5 - 1) * 2 = 4 + 8 = 12.