Sequences and Series

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What is a sequence?

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A sequence is an ordered list of numbers that follow a specific pattern or rule.

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What is a series?

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A series is the sum of the terms of a sequence.

Front

Define an arithmetic sequence.

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An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant.

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What is the formula for the nth term of an arithmetic sequence?

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The nth term (a_n) can be calculated using the formula: a_n = a_1 + (n-1)d, where a_1 is the first term and d is the common difference.

Front

Define 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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What is the formula for the nth term of a geometric sequence?

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The nth term (a_n) can be calculated using the formula: a_n = a_1 * r^(n-1), where a_1 is the first term and r is the common ratio.

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How do you find the sum of the first n terms of an arithmetic series?

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The sum (S_n) can be calculated using the formula: S_n = n/2 * (a_1 + a_n), where a_n is the nth term.

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How do you find the sum of the first n terms of a geometric series?

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The sum (S_n) can be calculated using the formula: S_n = a_1 * (1 - r^n) / (1 - r), for r ≠ 1.

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What is the binomial theorem?

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The binomial theorem describes the algebraic expansion of powers of a binomial. It states that (a + b)^n = Σ (n choose k) * a^(n-k) * b^k, where k = 0 to n.

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What does 'n choose k' mean in combinatorics?

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'n choose k' (denoted as C(n, k) or nCk) represents the number of ways to choose k elements from a set of n elements without regard to the order of selection.