Section 11.1 Sequences

Sequence – list of values following a pattern Arithmetic – from term to term there is a common difference we’ll call d Geometric – from term to term there is a common ratio we’ll call r Everything else

Definition (more mathematical) A sequence is a function whose domain is the set of positive integers. A sequence is a function (input then output), and it will have a graph. The positive integers are evaluated within the function to give us the terms of the sequence. Example:{a n } = { (n-1) / n }

Identify the first 6 terms of the sequence {a n } = { (n-1) / n } Calculator: seq(expression, variable, start, stop, increment) sequence located in LIST OPS 5

Identify the first 6 terms of the sequence {b n } = {(-1) n-1 ( 2 / n )} Calculator option 2: SEQ mode, “Y=“, nMin = 1 u(n)=expression, u(nMin)=2 Look at the table

Factorials The factorial symbol, n!, is defined as follows: 0! = 11! = 1 If n ≥ 2 is an integer n! = n(n-1)(n-2)... (3)(2)(1)n! = n(n-1)! MATH PRB 4

Examples Find: a) 5! b) 10! c) (4!)(6!)

RECURSIVE FORMULAS When the sequence is defined by the term(s) preceding the nth term Must be given one or more of the first few terms All other terms are then defined using the previous terms MOST FAMOUS Fibonacci Sequence a 0 = 1, a 1 = 1, a 2 = 2, a 3 = 3, a 4 = 5,…, a k = a k-2 + a k-1

Summation Notation Rather than write: a 1 + a 2 + a a n we express the sum using summation notation. n ∑a k k=1 n ∑a k = a 1 + a 2 + a a n k=1 The index tells you where to start and end (bottom to top), although we often use k, it doesn’t matter

Rewrite the following 5 A)∑ k -1 k=1 4 B)∑ k! k=1

Write using summation notation a) b) 1 + ½ + ¼ + 1/ /(2 n-1 )

Properties of Sequences If {a n } and {b n } are 2 sequences and c is a real number, then:

Find the sum of each sequence 5 A)∑ 3k k=1 4 B)∑ k 2 – 7k + 2 k=1

Things to watch for… (-1) n or (-1) n±1 when the sign changes each term (2n) and (2n ± 1) for even and odd If the terms differ by the same amount, think linear If the 2 nd level terms differ by one amount, think quadratic

Applications Annuity Formula A 0 = M (initial amount deposited) r = interest rate expressed as a percent in decimal form N: number of compound periods per year P: periodic deposit made at each payment period A n = amount after n deposits have been made

Applications Amortization Formula A 0 = B (initial amount borrowed) r = interest rate expressed as a percent in decimal form N: number of compound periods per year P: periodic deposit made at each payment period A n = amount after n payments have been made