Pre-Calculus Sequences and Series

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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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CCSS.HSF.BF.A.2

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

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The common ratio (r) can be found by dividing any term in the sequence by the previous term, i.e., r = a_n / a_(n-1).

Tags

CCSS.HSF.BF.A.2

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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 between consecutive terms is constant, known as the common difference.

Tags

CCSS.HSF.BF.A.2

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How do you find the common difference in an arithmetic sequence?

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The common difference (d) can be found by subtracting any term from the subsequent term, i.e., d = a_n - a_(n-1).

Tags

CCSS.HSF.BF.A.2

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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 of a geometric series can be calculated using the formula: S_n = a_1 * (1 - r^n) / (1 - r), where a_1 is the first term and r is the common ratio.

Tags

CCSS.HSA.SSE.B.4

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What is the formula for the sum of the first n terms of an arithmetic series?

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The sum S_n of the first n terms of an arithmetic series can be calculated using the formula: S_n = n/2 * (a_1 + a_n), where a_1 is the first term and a_n is the nth term.

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

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The first term (a_1) is crucial as it serves as the starting point for the series and is multiplied by the common ratio to generate subsequent terms.

Tags

CCSS.HSA.SSE.B.4

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What happens to the sum of a geometric series if the common ratio is greater than 1?

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If the common ratio is greater than 1, the sum of the series diverges to infinity as more terms are added.

Tags

CCSS.HSA.SSE.B.4

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What is the behavior of the sum of a geometric series if the common ratio is between 0 and 1?

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If the common ratio is between 0 and 1, the sum converges to a finite value as more terms are added.

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How do you determine if a series converges or diverges?

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A series converges if the sum approaches a finite limit as the number of terms increases; it diverges if the sum increases without bound.