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

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The nth term 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 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 identify the common difference in an arithmetic sequence?

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The common difference (d) in an arithmetic sequence can be found by subtracting any term from the subsequent term: d = a_(n+1) - a_n.

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How do you identify the common ratio in a geometric sequence?

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The common ratio (r) in a geometric sequence can be found by dividing any term by the previous term: r = a_n / a_(n-1).

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What is the recursive formula for an arithmetic sequence?

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The recursive formula for an arithmetic sequence is: a_n = a_(n-1) + d, where d is the common difference.

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What is the recursive formula for a geometric sequence?

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The recursive formula for a geometric sequence is: a_n = a_(n-1) * r, where r is the common ratio.

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What is the relationship between exponential functions and geometric sequences?

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Exponential functions are related to geometric sequences because they both involve a constant multiplicative factor; in geometric sequences, each term is a constant multiple of the previous term.

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How can you determine if a sequence is arithmetic or geometric?

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To determine if a sequence is arithmetic, check if the differences between consecutive terms are constant. To determine if it is geometric, check if the ratios of consecutive terms are constant.