Chapter 9-1 Introduction to Sequences
Obj: To find the nth term of a sequence, and to write rules for sequences.
Who uses this? – Sequences are used to model many natural phenomenon, such as changes in rabbit population
In 1202, Italian Mathematician Leonardo Fibonacci described how fast rabbits breed
under ideal circumstances. Fibonacci noted the number of pairs of pairs of rabbits each
month and formed a famous pattern called the Fibonacci sequence.
Sequences
A sequence is an ordered set of numbers. Each number in the sequence is called a term
of the sequence.
Infinite Sequence – continues without end, unlimited number of terms (ex. Natural
Numbers)
Finite Sequence – has a limited number of terms.
One way to consider a sequence is with sequential natural numbers as the Domain, and
the terms of the sequence as the Range.
Term number:
Term value:
1
1
2
5
3
9
4
13
5
17
6
7
8…
____ ____ ____
Values in the domain are called Term Numbers, and are represented by n. Instead of
functional notation, such as a(n), sequence values are written by using subscripts. The
first term is 𝑎1 , the second term is 𝑎2 , and the nth term is 𝑎𝑛 . Because a sequence is a
function, each number “n” has only one term value associated with 𝑎𝑛 .
Recursive Formula – is a rule in which one or more previous terms are used to generate
the next terms. (ex The Fibonacci Sequence is expressed by the rule:
𝑎1 = 1, 𝑎2 = 1,
𝑎𝑛𝑑 𝑎𝑛 = 𝑎𝑛−2 + 𝑎𝑛−1 ,
𝑤ℎ𝑒𝑟𝑒 𝑛 ≥ 3.
Ex 1. Finding Terms of a Sequence by using a Recursive Formula:
Find the first five terms of the sequence with:
a)
𝑎1 = 5 𝑎𝑛𝑑 𝑎𝑛 = 2𝑎𝑛−1 + 1 𝑓𝑜𝑟 𝑛 ≥ 2. (Hint: make a table)
𝑛
2𝑎𝑛−1 + 1
𝑎𝑛
1
Given
5
2
2(5) + 1
3
4
5
b)
𝑎1 = −3 𝑎𝑛𝑑 𝑎𝑛 = 𝑎𝑛−1 − 8
Ex 2. Finding Terms of a Sequence by Using an Explicit Formula
Find the first five terms of the sequence with:
𝑎) 𝑎𝑛 = 2𝑛 − 3
(again… a table helps)
𝑏) 𝑎𝑛 = 𝑛2 − 2𝑛
Writing Rules for sequences
TIP: Linear patterns – have constant first differences
Quadratic patterns – have constant second differences
Exponential patterns – have constant ratios
Write a possible explicit rule for the nth term of each sequence
𝑎) 7, 5, 3, 1, −1, …
Terms: 7
(Write the terms and try to find a common difference)
5
3
1
-1
The first differences are constant
so the sequence is linear.
You try…
𝑏) 25, 15, 5, −5, 15, …
Terms: 25
15
Common difference?
5
-5
-15
12
24
48
𝑏) 3, 6, 12, 24, 48, …
Terms: 3
6
Homework p.629/ 16-24 all
find common differences and ratios