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Section 9.1 – Sequences

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Sequence First Term Second Term n th Term Generally, we will concentrate on infinite sequences, that is, sequences with domains that are infinite subsets of the positive integers.

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Recursive Formula A formula that requires the previous term(s) in order to find the value of the next term. Example: Find a Recursive Formula for the sequence below. 2, 4, 8, 16, …

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Explicit Formula A formula that requires the number of the term in order to find the value of the next term. Example: Find an Explicit Formula for the sequence below. 2, 4, 8, 16, … The Explicit Formula is also known as the General or nth Term equation.

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A sequence which has a constant difference between terms. The rule is linear. Example: 1, 4, 7, 10, 13,… (generator is +3) Arithmetic Sequences na(n) Discrete Explicit Formula Recursive Formula

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Write an equation for the nth term of the sequence: a(0) is not in the sequence! Do not include it in tables or graphs! White Board Challenge 36, 32, 28, 24, … n=1 n=2 n=3 n=4 40, n=0 – 4 Sequences typically start with n=1 First find the generator Then find the n=0 term.

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A sequence which has a constant ratio between terms. The rule is exponential. Example: 4, 8, 16, 32, 64, … (generator is x2) Geometric Sequences nt(n) x2 Discrete Explicit Formula Recursive Formula

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Write an equation for the nth term of the sequence: White Board Challenge n=1 n=2 n=3 n=4 n=0 3, 15, 75, 375, … x5 a(0) is not in the sequence! Do not include it in tables or graphs! Sequences typically start with n=1 First find the generator Then find the n=0 term.

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New Sequences n=1 n=2 n=3 n=4 n=5

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White Board Challenge n=1 n=2 n=3 n=4 n=5

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Monotonic Sequence

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Example 2 IF:THEN: Since the denominator is smaller: OR

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Bounded Sequence

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Example

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Limit of a Sequence

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Reminder: Properties of Limits Let b and c be real numbers, let n be a positive integer, and let f and g be functions with the following limits: Constant Function Limit of x Limit of a Power of x Scalar Multiple

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Reminder: Properties of Limits Let b and c be real numbers, let n be a positive integer, and let f and g be functions with the following limits: Sum/Difference Product Quotient Power

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Example Converges to 1 Diverges Converges to 0

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White Board Challenge Converges to

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Absolute Value Theorem

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Example Because of the Absolute Value Theorem, Converges to 0 Since the limit does not equal 0, we can not apply the Absolute Value Theorem. It does not mean it diverges. Another test is needed. The sequence diverges since it does not have a limit: -1,1,-1,1,-1,…

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Theorem: Bounded, Monotonic Sequences The sequence appears to be monotonic: It is increasing. The limit of the sequence appears to be 6. Since the sequence appears to be monotonic and bounded, it appears to converge to 6.

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