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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What is the explicit formula for an arithmetic sequence?

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The explicit formula for an arithmetic sequence is given by: 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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What is the explicit formula for a geometric sequence?

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The explicit formula for a geometric sequence is given by: 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 does 'r' represent in a geometric sequence?

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In a geometric sequence, 'r' represents the common ratio, which is the factor by which we multiply each term to get the next term.

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How do you find the 9th term of a geometric sequence with a first term of 3 and a common ratio of 2?

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To find the 9th term, use the formula: a_n = a_1 * r^(n-1). Here, a_9 = 3 * 2^(9-1) = 3 * 256 = 768.

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What is the common difference in the arithmetic sequence 2, 5, 8, 11...?

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The common difference in this arithmetic sequence is 3, as each term increases by 3.

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

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Using the formula a_n = a_1 * r^(n-1), the 4th term is a_4 = 4 * 3^(4-1) = 4 * 27 = 108.

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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 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), for r ≠ 1.