Warm Up Some sequences follow predictable patterns, though the pattern might not be immediately apparent. Other sequences have no pattern at all. Explain, when possible, patterns in the following sequences:   a. 5, 4, 3, 2, 1 b. 3, 5, 1, 2, 4 c. 2, 4, 3, 5, 1, 5, 1, 5, 1, 5, 1, 7 d. S, M, T, W, T, F, S e. 31, 28, 31, 30, 31, 30, 31, 31, 30, 31, 30, 31 f. 1, 2, 3, 4, 5, ..., 999, 1000 g. 1, -1, 1, -1, 1, -1, ... h. 4, 7, 10, 13, 16, ... i. 10, 100, 1000, 10000, 100000, ...

Keeper GSE Accelerated Pre-Calculus Sequences and Series Keeper GSE Accelerated Pre-Calculus

Sequences A sequence is a function, where the domain is a set of consecutive positive integers beginning with 1. An infinite sequence is a function having for its domain the set of positive integers, {1, 2, 3, 4, 5, …}. A finite sequence is a function having for its domain a set of positive integers, {1, 2, 3, 4, 5, …, n}, for some positive integer n.

Sequence Formulas In a formula, the function values are known as terms of the sequence. The first term in a sequence is denoted as a1, the fifth term as a5 , and the nth term, or the general term, as an.

Finding the General Term Example: Predict the general term of the sequence 4, 16, 64, 256, …

Example Find the first 4 terms and the 9th term of the sequence whose general term is given by an = 4(2)n.

Alternating Sequence The power (2)n causes the sign of the terms to alternate between positive and negative, depending on whether the n is even or odd. This kind of sequence is called an alternating sequence.

Sums and Series

Example For the sequence 1, 3, 5, 7, 9, 11, 13, … find each of the following: a) S1 b) S5 c) S7

Example Find and evaluate the sum.

Write sigma notation for the sum 5 + 25 + 125 + … Example Write sigma notation for the sum 5 + 25 + 125 + …

Convergent and Divergent Sequences If a sequence has a limit such that the terms approach a unique number, then it is said to converge. If not, the sequence is said to diverge. Example: Determine whether each sequence is convergent or divergent.

Recursive Definitions A sequence may be defined recursively or by using a recursion formula. Such a definition lists the first term, or the first few terms, and then describes how to determine the remaining terms from the given terms.

Example Example: Find the first 4 terms of the sequence defined by a1 = 3, an= 3an-1  2 for n  1.

Real World Example Suppose that when a plant first starts to grow, the stem has to grow for two months before it is strong enough to support branches. At the end of the second month, it sprouts a new branch and will continue to sprout one new branch each month. The new branches also each grow for two months and then start to sprout one new branch each month. If this pattern continues, how many branches will the plant have after 10 months?

Answer to Tree problem…