Geometric Series Practice

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What is the formula for the sum of a finite geometric series?

Back

The sum of a finite geometric series is given by S_n = \frac{a_1(1 - r^n)}{1 - r}, where S_n is the sum, a_1 is the first term, r is the common ratio, and n is the number of terms.

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What does it mean for a geometric series to converge?

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A geometric series converges if the absolute value of the common ratio r is less than 1 (|r| < 1). In this case, the series approaches a finite limit.

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What does it mean for a geometric series to diverge?

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A geometric series diverges if the absolute value of the common ratio r is greater than or equal to 1 (|r| ≥ 1). In this case, the series does not approach a finite limit.

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How do you determine if a geometric series converges or diverges?

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To determine if a geometric series converges or diverges, check the common ratio r. If |r| < 1, it converges; if |r| ≥ 1, it diverges.

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If a ball is dropped from a height of 25 cm and bounces to 80% of its previous height, what is the height after the first bounce?

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The height after the first bounce is 25 cm * 0.8 = 20 cm.

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What is the total distance traveled by a bouncing ball after n bounces if it bounces to a constant fraction of its previous height?

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The total distance traveled can be calculated by summing the distances of the downward and upward bounces, which forms a geometric series.

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What is the common ratio in a geometric series?

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The common ratio in a geometric series is the factor by which each term is multiplied to get the next term. It is denoted as r.

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What is the first term of a geometric series?

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The first term of a geometric series is the initial value from which the series begins, denoted as a_1.

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How do you calculate the total distance traveled by a ball that bounces multiple times?

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To calculate the total distance, sum the distances of all downward and upward bounces, which can be expressed as a geometric series.

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What is the significance of the sum formula in geometric series?

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The sum formula allows us to calculate the total sum of a finite geometric series without having to add each term individually.