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What is a geometric progression?
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A geometric progression is 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 (r) in a geometric progression calculated?
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The common ratio is calculated by dividing any term in the sequence by the previous term.
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What is the formula for the nth term of a geometric progression?
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The nth term (a_n) is given by the formula a_n = a * r^(n-1), where a is the first term and r is the common ratio.
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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 found by multiplying the previous one by a fixed number, while in an arithmetic sequence, a fixed number is added to the previous term.
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What are finite and infinite geometric sequences?
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A finite geometric sequence has a limited number of terms, while an infinite geometric sequence continues indefinitely.
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Can a geometric sequence have negative terms?
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Yes, a geometric sequence can have negative terms if the common ratio is negative.
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What happens if the common ratio in a geometric progression is equal to 1?
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If the common ratio is 1, all terms in the sequence will be the same as the first term.
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Give an example of a geometric progression with a common ratio of 2.
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An example is the sequence 3, 6, 12, 24, where the first term is 3 and the common ratio is 2.
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What is a geometric series?
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A geometric series is the sum of the terms of a geometric sequence.
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How does the sum of an infinite geometric series differ from a finite one?
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An infinite geometric series can converge to a finite sum if the common ratio is between -1 and 1, whereas a finite series always has a definite sum.
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What is the formula for the sum of the first n terms of a geometric series?
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The sum S_n of the first n terms is S_n = a(1-r^n)/(1-r), where a is the first term and r is the common ratio.
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What condition must a geometric series meet to converge?
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For an infinite geometric series to converge, the absolute value of the common ratio must be less than 1 (|r| < 1).
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How can geometric progressions be applied in finance?
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Geometric progressions are used to model compound interest and exponential growth in financial calculations.
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What is the significance of the zero term in a geometric sequence?
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In a geometric sequence, having a zero term means that subsequent terms will all be zero if the common ratio is non-zero.
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In a geometric sequence with a common ratio of 1/2 and a first term of 8, what is the 4th term?
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The 4th term is 8 * (1/2)^(4-1) = 8 * 1/8 = 1.
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