2.5 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.

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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.

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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 term that follows it: d = a(n) - a(n-1).

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

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The common ratio (r) can be found by dividing any term by the term that precedes it: r = a(n) / a(n-1).

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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 found using the formula: a(n) = a(1) + d(n-1), 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 found 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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Given the first term 5 and common difference 3, what is the 5th term of the arithmetic sequence?

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The 5th term is 5 + 3(5-1) = 5 + 12 = 17.

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Given the first term 2 and common ratio 3, what is the 4th term of the geometric sequence?

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The 4th term is 2 * 3^(4-1) = 2 * 27 = 54.

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Identify whether the sequence 3, 6, 12, 24,... is Arithmetic or Geometric and find the common ratio or difference.

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The sequence is Geometric with a common ratio of 2.

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Identify whether the sequence 10, 7, 4, 1,... is Arithmetic or Geometric and find the common difference or ratio.

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The sequence is Arithmetic with a common difference of -3.