Arithmetic and Geometric 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 a geometric series?

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A geometric series is the sum of the terms of a geometric sequence, where each term after the first is obtained by multiplying the previous term by a constant ratio.

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How do you find the sum of the first n terms of an arithmetic series?

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The sum of the first n terms (S_n) of an arithmetic series can be calculated using the formula: S_n = n/2 * (a + l), where a is the first term, l is the last term, and n is the number of terms.

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How do you find the sum of the first n terms of a geometric series?

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The sum of the first n terms (S_n) of a geometric series can be calculated using the formula: S_n = a * (1 - r^n) / (1 - r), where a is the first term, r is the common ratio, and n is the number of terms.

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What is the common difference in an arithmetic series?

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The common difference is the constant amount that is added to each term to get the next term in an arithmetic sequence.

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What is the common ratio in a geometric series?

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The common ratio is the constant factor by which each term is multiplied to get the next term in a geometric sequence.

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Calculate the sum of the first 6 terms of the geometric series: -4/5, 8, -80, 800, ...

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Find the sum of the first 25 terms of the series: 6 + 14 + 22 + 30 + ...

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S_{25} = 2550.

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If Mike's starting salary is $35,000 and increases by $1,200 each year, what is his total salary after 10 years?

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Total salary = $404,000.

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Find the sum of the first 718 terms of the series: 2 + 4 + 6 + ... + 1436.

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S_{718} = 516,242.