Arithmetic and Geometric Sequences

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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 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 get the next term.

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How can you determine if a sequence is arithmetic?

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A sequence is arithmetic if the difference between consecutive terms is constant. You can check this by subtracting each term from the next.

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How can you determine if a sequence is geometric?

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A sequence is geometric if the ratio of consecutive terms is constant. You can check this by dividing each term by the previous term.

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What is the formula to find the sum of the first n terms of an arithmetic sequence?

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The sum (S_n) of the first n terms of an arithmetic sequence 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 to find the sum of the first n terms of a geometric sequence?

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The sum (S_n) of the first n terms of a geometric sequence can be calculated using the formula: S_n = a_1 * (1 - r^n) / (1 - r), for r ≠ 1.