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 = 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 * (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 calculate the total earnings over a period with a constant raise?

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Total earnings can be calculated using the formula: Total = (initial salary * number of years) + (raise * (number of years * (number of years + 1) / 2)).

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What is the common difference in an arithmetic sequence?

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

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The nth term can be found using the formula: a_n = a + (n - 1)d, where a is the first term, d is the common difference, and n is the term number.

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What is the significance of the first term in a series?

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The first term is crucial as it serves as the starting point for calculating subsequent terms and the sum of the series.

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How do you calculate the total cost with a percentage increase each year?

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Total cost can be calculated using the formula: Total = initial cost * (1 + rate)^n, where rate is the percentage increase expressed as a decimal and n is the number of years.

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What is the formula for the sum of an arithmetic series when the first term and the common difference are known?

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The formula is S_n = n/2 * (2a + (n - 1)d), where a is the first term, d is the common difference, and n is the number of terms.