HSF-BF.A.2 Review 3

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What is a recursive formula?

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A recursive formula defines each term of a sequence using the previous term(s). For example, in the sequence defined by a_n = a_{n-1} + 5, each term is found by adding 5 to the previous term.

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How do you find the first four terms of a recursive sequence?

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To find the first four terms, start with the initial term and apply the recursive formula repeatedly. For example, if a_1 = -16 and a_n = a_{n-1} + 5, the terms are: a_1 = -16, a_2 = -11, a_3 = -6, a_4 = -1.

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What is an explicit function?

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An explicit function provides a direct formula to calculate the nth term of a sequence without needing previous terms. For example, f(n) = 2 * 4^{n-1} is an explicit function for a sequence.

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What is the difference between recursive and explicit formulas?

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A recursive formula defines terms based on previous terms, while an explicit formula allows direct calculation of any term in the sequence.

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How do you evaluate an exponential function?

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To evaluate an exponential function like g(n) = (-3)^n, substitute the value of n into the function. For example, g(4) = (-3)^4 = 81.

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What is the general form of a recursive formula for an arithmetic sequence?

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The general form is a_n = a_{n-1} + d, where d is the common difference between terms.

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

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The common difference can be found by subtracting any term from the subsequent term. For example, in the sequence 6, 17, 28, 39, 50, the common difference is 11.

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

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The nth term 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.

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

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The nth term 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 derive the explicit formula from a recursive formula?

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To derive the explicit formula, identify the pattern in the recursive formula and express it in terms of n. For example, from f(n) = f(n-1) * 4, we can derive f(n) = 2 * 4^{(n-1)}.