Geometric Progression Fundamentals

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What is a geometric progression?
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A sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio.
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How is the common ratio in a geometric sequence determined?
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By dividing any term by the previous term.
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Write the formula for the n-th term of a geometric sequence.
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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 sum of the first n terms of a geometric series?
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S_n = a * (1 - r^n) / (1 - r) for r ≠ 1.
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In a geometric sequence, if the first term is 3 and the common ratio is 2, what is the fourth term?
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The fourth term is 3 * 2^3 = 24.
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How does a geometric sequence differ from an arithmetic sequence?
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In a geometric sequence, each term is multiplied by a constant ratio, whereas in an arithmetic sequence, a constant is added to each term.
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What is the common ratio if the sequence is 5, 15, 45?
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The common ratio is 15/5 = 3.
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Explain what happens if the common ratio is 1.
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The sequence remains constant, with each term equal to the first term.
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Given the sequence 1, 1/2, 1/4, what is the common ratio?
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The common ratio is 1/2.
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What is an infinite geometric series?
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A geometric series that has an infinite number of terms.
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For an infinite geometric series, when does it converge?
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It converges when the absolute value of the common ratio is less than 1.
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What is the formula for the sum of an infinite geometric series?
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S = a / (1 - r) for |r| < 1.
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Identify the first term and common ratio of the sequence 10, -20, 40, -80.
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First term is 10, common ratio is -2.
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If the 6th term of a geometric sequence is 128 and the common ratio is 2, what is the first term?
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First term a_1 = 128 / 2^(6-1) = 4.
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True or False? The sequence 1/3, 1, 3, 9 is geometric.
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True, with a common ratio of 3.
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