Arithmetic Series

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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, where each term after the first is obtained by adding a constant difference to the previous term.

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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 formula is: S_n = \frac{n}{2} (a_1 + a_n) or S_n = \frac{n}{2} (2a_1 + (n-1)d), where S_n is the sum, a_1 is the first term, a_n is the nth term, d is the common difference, and n is the number of terms.

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

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The common difference (d) is found by subtracting the first term from the second term: d = a_2 - a_1.

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If a_1 = 2 and d = 3, what is the 5th term of the arithmetic sequence?

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The 5th term can be calculated using the formula: a_n = a_1 + (n-1)d. Thus, a_5 = 2 + (5-1)3 = 14.

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What is the 6th partial sum of the series 2 + 6 + 10 + 14 + ...?

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The 6th partial sum S_6 = \frac{n}{2} (a_1 + a_n) = \frac{6}{2} (2 + 26) = 84.

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Given a_1 = -3, d = 2, and S_n = 21, how many terms does the arithmetic sequence have?

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Using the formula S_n = \frac{n}{2} (2a_1 + (n-1)d), we can solve for n. The answer is 7.

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Evaluate the arithmetic series described: a_1 = 5, a_n = 145, Find S_{15}.

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Using the formula S_n = \frac{n}{2} (a_1 + a_n), we find S_{15} = \frac{15}{2} (5 + 145) = 1125.

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What is the nth term formula for an arithmetic sequence?

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The nth term formula is: a_n = a_1 + (n-1)d.

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How do you determine the number of terms in an arithmetic series if you know the first term, common difference, and the last term?

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You can use the formula: n = \frac{(a_n - a_1)}{d} + 1.