Finite Geometric Series

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What is a finite geometric series?

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A finite geometric series is the sum of the terms of a geometric sequence that has a finite number of terms.

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

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The sum S of the first n terms of a finite geometric series can be calculated using the formula: S = a(1 - r^n) / (1 - r), where a is the first term and r is the common ratio.

Front

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.

Front

How do you find the common ratio in a geometric series?

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The common ratio r can be found by dividing any term by the previous term in the series.

Front

If a geometric series starts with 4 and has a common ratio of 1/2, what are the first five terms?

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The first five terms are 4, 2, 1, 1/2, 1/4.

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What is the sum of the first five terms of the series 4, 2, 1, 1/2, 1/4?

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The sum is 4 + 2 + 1 + 1/2 + 1/4 = 7.75.

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What happens to the sum of a finite geometric series as the number of terms increases?

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As the number of terms increases, the sum approaches a limit if the common ratio is between -1 and 1.

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What is the significance of the first term in a geometric series?

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The first term (a) is crucial as it determines the starting point of the series and influences the overall sum.

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

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A geometric series converges if the absolute value of the common ratio |r| < 1; it diverges if |r| ≥ 1.

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What is the sum of the first n terms of the geometric series 4, -12, 36, -108?

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The sum of the first n terms can be calculated using the formula for the sum of a finite geometric series.