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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 (d).

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How do you find 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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What is a geometric sequence?

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A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio (r).

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How do you find the nth term of a geometric sequence?

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The nth term (a_n) of a geometric sequence can be calculated using the formula: a_n = a_1 * r^(n-1), where a_1 is the first term and r is the common ratio.

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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_n is the nth term.

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What is the formula for the sum of an infinite geometric series?

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The sum (S) of an infinite geometric series exists if the absolute value of the common ratio (|r|) is less than 1, and is given by: S = a_1 / (1 - r), where a_1 is the first term.

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What is the common difference in an arithmetic sequence?

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The common difference (d) in an arithmetic sequence is the constant amount that each term increases or decreases by from the previous term.

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What is the common ratio in a geometric sequence?

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The common ratio (r) in a geometric sequence is the factor by which each term is multiplied to obtain the next term.

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How do you determine if a series converges or diverges?

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A series converges if the sum approaches a finite limit as more terms are added. A series diverges if the sum increases without bound or does not approach a finite limit.

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What is the explicit formula for the nth term of the sequence 25, 21, 17, 13,...?

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The explicit formula for the sequence is a_n = -4n + 29.