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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 of any two successive members is a constant.
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What is the formula to find the nth term in 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 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.
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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 and r is the common ratio.
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What is the common ratio in the geometric sequence 3, 9, 27, 81?
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The common ratio is 3.
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What is a Fibonacci sequence?
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The Fibonacci sequence is a series of numbers where each number is the sum of the two preceding ones, usually starting with 0 and 1.
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What are the first five terms of the Fibonacci sequence?
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0, 1, 1, 2, 3
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How do you determine if a sequence is arithmetic?
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Check if the difference between consecutive terms is always the same. If it is, the sequence is arithmetic.
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How can you check if a sequence is geometric?
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Divide each term by the previous term to check if the ratio is constant. If it is, the sequence is geometric.
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What is the sum formula for the first n terms of an arithmetic sequence?
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The sum S_n of the first n terms is given by the formula: S_n = n/2 * (a_1 + a_n).
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How do you calculate the sum of a finite geometric sequence?
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The sum S of the first n terms of a geometric sequence is calculated using: S_n = a_1 * (1-r^n) / (1-r), where r ≠ 1.
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What is a sequence, in mathematical terms?
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A sequence is an ordered list of numbers following a particular pattern.
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Why is understanding sequences important in mathematics?
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Sequences are crucial in mathematics for modeling and solving real-world problems involving patterns and progressions, like population growth and interest rates.
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What is a recursive formula in sequences?
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A recursive formula defines each term of a sequence using the previous terms.
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Give an example of a recursive formula.
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An example is the Fibonacci sequence: F(n) = F(n-1) + F(n-2), with seed values F(0) = 0 and F(1) = 1.
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