Sequences Flashcard

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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. This difference is called the common difference.

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How do you find the nth term of an arithmetic sequence?

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The nth term of an arithmetic sequence can be found using the formula: \( a_n = a_1 + (n-1)d \), where \( a_1 \) is the first term, \( d \) is the common difference, and \( n \) is the term number.

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

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The sum of the first n terms of an arithmetic sequence is given by: \( S_n = \frac{n}{2} (a_1 + a_n) \) or \( S_n = \frac{n}{2} (2a_1 + (n-1)d) \).

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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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How do you find the nth term of a geometric sequence?

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The nth term of a geometric sequence can be found using the formula: \( a_n = a_1 \cdot r^{(n-1)} \), where \( a_1 \) is the first term, \( r \) is the common ratio, and \( n \) is the term number.

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What is the formula for the sum of the first n terms of a geometric sequence?

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The sum of the first n terms of a geometric sequence is given by: \( S_n = a_1 \frac{1 - r^n}{1 - r} \) (for \( r \neq 1 \)).

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What is sigma notation?

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Sigma notation is a way to represent the sum of a sequence of terms using the Greek letter sigma (\( \Sigma \)). It allows for concise representation of sums.

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How do you evaluate a sum using sigma notation?

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To evaluate a sum using sigma notation, substitute the values of the index into the expression and sum the results.

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What is the common difference in an arithmetic sequence?

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The common difference in an arithmetic sequence is the constant amount that each term increases or decreases by from the previous term.

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What is the common ratio in a geometric sequence?

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The common ratio in a geometric sequence is the constant factor by which each term is multiplied to get the next term.