13.1 Sequences

A sequence is a function whose domain is the set of positive integers or a subset of the positive integers that consists of the n integers 1, 2, 3, … n.In other words… a list of numbers that follow a patternThere are all kinds of sequences

2, 4, 6, 8 1, 1, 2, 3, 5, 8, 13, 21, …

A sequence can be finite (a particular number of terms) or infinite (go on forever)

Ex 1) Find the first five terms of the infinite sequence defined by f (n) = 2n + 3 f (1) = 5 f (2) = 7 f (3) = 9 f (4) = 11 f (5) = 13

So, 5, 7, 9, 11, 13, …

these are called terms

1 1 1 1 1 1 11, , , ,... 1, , , ,

3 9 27 2 3 4 5

We can also use an to denote a specific term such as a1, a2, or a10.The general term, an, is the nth term.The formula used is called the rule of the sequence.

Ex 2) Find the nineteenth term of the sequence given by the rule bn = 3n – 1 b19 = 319 – 1 = 1, 162, 261, 466

A rule like in Ex 2 is called an explicit formula – it gives the nth term as a function of n.

Ex 3) Find an explicit formula for:a) 10, 20, 30, 40, … 10n

b) 2 2 2 2 2

, , , ,...3 9 27 81 3n

Another way to give a rule for a sequence is using a recursive formula –it gives an initial term and then defines an using a preceding term.Classic Example – The Fibonacci Sequence

a1 = 1, a2 = 1, an = an – 1 + an – 2, n > 2

Ex 4) Find the next four terms of the sequence given by the recursive formula a1 = 3 and an = 2an – 1 – 1, n > 1

5, 9, 17, 33

A sequence can be defined by either formula, recursive or explicit.

Ex 5) Find an explicit formula for the sequence defined recursively by a1 = 7 and an = an – 1 + 3, n > 1Write a few terms: 7, 10, 13, 16, 19

an = 4 + 3n

We can also, if given enough terms, discover a recursive formula.

Ex 6) Mrs. Fox decided to ease into saving her tutoring money. The first week, she set aside $1, the second week $2, the third week $4, the fourth week $7 and so on. It generated the sequence 1, 2, 4, 7, 11, 16, 22, … Find a recursive formula for the sequence.

thoughts…. First add 1, then add 2, then add 3, …a1 = 1an = an – 1 + (n – 1), n > 1

Homework

#1301 Pg 677 #1, 5, 7, 9, 15, 20, 23, 25, 27, 33, 35, 38, 39