Arithmetic and Geometric Sequences and Series Review

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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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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 common difference in an arithmetic sequence?

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The common difference (d) can be found by subtracting any term from the subsequent term: d = a(n+1) - a(n).

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What is the formula for the nth term of an arithmetic sequence?

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The nth term (a_n) of an arithmetic sequence can be calculated 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 the formula for 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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How do you evaluate the sum of a finite geometric series?

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The sum (S_n) of the first n terms of a geometric series can be calculated using the formula: S_n = a_1 * (1 - r^n) / (1 - r), 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 an infinite geometric series?

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The sum (S) of an infinite geometric series can be calculated using the formula: S = a_1 / (1 - r), where |r| < 1.

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Given a sequence, how can you determine if it is arithmetic or geometric?

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To determine if a sequence is arithmetic, check if the difference between consecutive terms is constant. To determine if it is geometric, check if the ratio of consecutive terms is constant.

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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 found by dividing any term by the previous term: r = a(n) / a(n-1).

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How do you find the 5th term of an arithmetic sequence?

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To find the 5th term (a_5) of an arithmetic sequence, use the formula: a_5 = a_1 + 4d, where a_1 is the first term and d is the common difference.