Section 8.1 Sequences

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Section 8.1 Sequences Learning Targets: I can identify and use arithmetic and geometric sequences I can graph a sequence I can find the limits of sequences (arithmetic and geometric)

Defining a Sequence A sequence an is a list of numbers written in an explicit order. If the domain is finite, then the sequence is a finite sequence. If the domain is an infinite subset of positive integers, it is an infinite sequence.

Example 1 Find the first six terms and the 100th term of the sequence:

Definitions An explicit formula is defined in terms of n. A recursive formula relates each new term to a previous term.

Example 2 Find the first four terms and the eighth term for the sequence defined recursively by the conditions: for all n 2 2 options… Keep adding until the 8th term. Write an explicit definition to find the 8th term (this isn’t ALWAYS an option).

Arithmetic and Geometric Sequences

Example 3 For each of the following arithmetic sequences, find (a) the common difference, (b) a recursive rule for the nth term, (c) an explicit rule for the nth term, and (d) the ninth term. Sequence 1: -5, -2, 1, 4, 7, … Sequence 2: ln 2, ln 6, ln 18, ln 54, …

Definition

Example 4 For each of the following geometric sequences, find (a) the common ratio, (b) a recursive rule for the nth term, (c) an explicit rule for the nth term, and (d) the tenth term. Sequence 1: 1, -2, 4, -8, 16, … Sequence 2: 10-2, 10-1, 1, 10, 102, …

8.1A Homework #5, 9, 11, 15, 19, 21

Please have your homework out so I can come around to check it off!

Example 5 The second and fifth terms of a geometric sequence are 6 and -48 respectively. Find the first term, the common ratio, and an explicit rule for the nth term. -48 6

Limit of a Sequence Definition: We write and say that the sequence converges to L. Sequences that do not have limits diverge. Sum Rule: Difference Rule: Product Rule: Constant Multiple Rule: Quotient Rule

Therefore, the sequence CONVERGES. Example 6 Determine whether the sequence converges or diverges. If it converges, find its limit. Therefore, the sequence CONVERGES.

Example 7 2 limits, therefore DIVERGES Arithmetic… so always DIVERGES Determine whether the sequence with given nth term converges or diverges. If it converges, find the limit. (a) n=1, 2, … (b) For all n 2 2 limits, therefore DIVERGES Arithmetic… so always DIVERGES

Sandwich (“Squeeze”) Theorem Min value of cos n = -1 Max value of cos n = 1

Absolute Value Theorem

8.1B Homework #31 – 37 odd, 41, 49 - 54