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 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 + l), where S_n is the sum of the first n terms, a is the first term, l is the last term, and n is the number of terms.

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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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What is the formula for the sum of the first n terms of a geometric series?

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The formula is: S_n = a \frac{(1 - r^n)}{(1 - r)}, where S_n is the sum of the first n terms, a is the first term, r is the common ratio, and n is the number of terms.

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How do you find the total earnings of an employee with a fixed raise each year?

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Use the formula for the sum of an arithmetic series: S_n = \frac{n}{2} (2a + (n-1)d), where a is the initial salary, d is the raise, and n is the number of years.

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Calculate the total earnings over 5 years for a salary of $48,000 with a $1,340 raise each year.

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Total earnings = $48,000 + $49,340 + $50,680 + $52,020 + $53,360 = $253,400.

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What is the common difference in an arithmetic series?

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The common difference is the constant amount added to each term to get the next term in the series.

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How do you find the sum of the first n even numbers?

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The sum of the first n even numbers can be calculated using the formula: S_n = n(n + 1).

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What is the common ratio in a geometric series?

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The common ratio is the factor by which each term in the series is multiplied to obtain the next term.

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

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The sum is 516,242.