Arithmetic & 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. For example, 2, 4, 6, 8 is an arithmetic sequence with a common difference of 2.

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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. For example, 3, 6, 12, 24 is a geometric sequence with a common ratio of 2.

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

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The nth term of an arithmetic sequence can be found using the formula: a_n = a_1 + (n-1)d, where a_1 is the first term, d is the common difference, and n is the term number.

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

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

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What is the common difference in an arithmetic sequence?

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The common difference is the fixed amount that each term increases or decreases by in an arithmetic sequence. For example, in the sequence 5, 8, 11, the common difference is 3.

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What is the common ratio in a geometric sequence?

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The common ratio is the fixed number that each term is multiplied by to get the next term in a geometric sequence. For example, in the sequence 2, 6, 18, the common ratio is 3.

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Can an arithmetic sequence have a common difference of zero?

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Yes, an arithmetic sequence can have a common difference of zero, which means all terms are the same. For example, 4, 4, 4, 4 is an arithmetic sequence with a common difference of 0.

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Can a geometric sequence have a common ratio of zero?

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No, a geometric sequence cannot have a common ratio of zero, as this would make all terms after the first term equal to zero.

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

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The sum of the first n terms of an arithmetic sequence can be calculated using the formula: S_n = n/2 * (a_1 + a_n), where S_n is the sum, a_1 is the first term, and a_n is the nth term.

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What is the sum of the first n terms of a geometric sequence?

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