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Sequences and Series (T) Students will know the form of an Arithmetic sequence. Arithmetic Sequence: There exists a common difference (d) between each term. Ex: 2, 6, 10, 14,…d = ? Ex: 17, 10, 3, -4, -11,…d = ? Ex: a, a+d, a+2d, a+3d, a+4d,… General term:

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(T) Students will know the form of a geometric sequence. Geometric Sequence: There exists a common ratio (r) between each term. Ex: 1, 3, 9, 27, 81 r = ? Ex: 64, -32, 16, -8, 4 r = ? Ex: a, ar, ar^2, ar^3, ar^4,… General term:

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(T) Students know the recursive definition. Recursive definitions: The next value of the sequence is determined using the previous term. Ex: Explicit definitions were given in the previous section.

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Ex: 23, 20, 17, 14,… Recursive def: Explicit def:

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(T) Students understand the concept of a series. Arithmetic and Geometric Series: A series is the sum of the terms of a sequence. Finite sum of an arithmetic series: Illustrate proof:

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(T) Students can find the sum of a series. Sum of a finite geometric series: Where r can not equal 1. Illustrate proof.

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(T) Students understand the concept of limits. Infinite Sequences: A sequence that continues forever. Ex: ½, ¼, 1/8, 1/16, …, (1/2)^n,… Limits: The sequence approaches some number but never reaches it. On a graph it is an asymptote. Ex:

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Theorem: (T) Students know the formula for an infinite geom. Series. Sum of an infinite geometric series:

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(T) Students can use mathematical induction for proofs. Mathematical Induction: Let S be a statement in terms of a positive integer n. Show that S is true for n=1 Assume that S is true for n=k, where is a positive integer, and then prove that S must be true for n = k + 1. Prove that Prove that n^3 + 2n is a multiple of 3.

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