Download Lesson 3.9 – Intro to Sequences ppt

UNIT 3 PART C:
ARITHMETIC &
GEOMETRIC SEQUENCES
ESSENTIAL QUESTION: WHAT ARE ARITHMETIC
& GEOMETRIC SEQUENCES AND WHEN DO WE
USE THEM?
F.IF.3: RECOGNIZE THAT SEQUENCES ARE
FUNCTIONS, SOMETIMES DEFINED
RECURSIVELY, WHOSE DOMAIN IS A SUBSET OF
THE INTEGERS.
INTRODUCTION
Sequence: a list of numbers
or shapes
• There are two types of
sequences: infinite and finite.
• An infinite sequence has an
infinite number of terms.
• A finite sequence ends and
has a certain number of terms.
• Examples:
•
•
•
An = 1,2, 3, 4,5,...
Un =
Term: the position of numbers
of shapes in a sequence
• Examples:
•
• U3
•
(infinite)
Bn = 2, 4,6,8,10. (finite)
A1 = 1
=
B4 = 8
SEQUENCES
• Sequences are ordered lists determined by
functions.
• The domain of the function that generates a
sequences is all natural numbers.
• A sequence is itself a function.
• We will work with two types of sequences:
Arithmetic and Geometric.
• There are two ways sequences are generally
defined – recursively and explicitly.
ARITHMETIC SEQUENCES
• Sequences whose terms increase or decrease by
the same amount.
• The amount of increase or decrease is called the
common difference.
• To determine the common difference, subtract the
second term from the first term. Then subtract the
third term from the second term and so on.
• Example:
• An = 1, 3,5, 7,9,... What is the common difference?
• Now you make your own example.
YOU TRY!
An = 2, 4,6,8,...
1. What is the common difference of the arithmetic
sequence?
1. What would the next term of the arithmetic
sequence above be?
RECURSIVE FORMULA
• Recursive formula is a formula that uses the previous
term to find the value of the next term.
• If the sequence is defined with a recursive formula,
the next term is based on the term before it and the
common difference.
• The recursive formula is known symbolically as
• The general form for Arithmetic Sequence is:
• Examples:
An = 1, 3,5, 7,9,...
d = 2, so we can write Anas
An = An-1 + 2
An-1
An = An-1 ± d
EXPLICIT FORMULAS
• Explicit formulas do not require previous terms to
find values of other terms.
• Explicitly defined sequences provide the function
that will generate each term.
• An = 2n + 3 or Bn = 5(3)n
• For arithmetic sequences, we define explicit
formulas similarly to linear functions.
• Why do you think we can use linear functions to help us
define arithmetic sequences?
• The formula is represented symbolically as:
• An = d(n -1) + A1 , d = common difference, n stays n,
A1 is the value of the first term.
GEOMETRIC SEQUENCES
• Geometric Sequences increase or decrease by a
common factor or ration, r.
• Dilations are examples of geometric sequences.
In this example, the circle is growing by
a factor of 2, meaning it is doubling in
size each term.
•
• More examples:
Un = 27,9,3,1,...
This sequence is decaying (decreasing)
at a factor of r = 1/3.
An = 2, 4,8,16, 32,... This sequence is growing (increasing) at
at a factor of r = 2.
YOU TRY!
1. What is the growth or decay factor (r) of
An = 16,8, 4,2,... ?
2. What is the growth or decay factor (r) of
Bn = 1, 3,9,27,.... ?
RECURSIVE FORMULA
• We can define geometric sequences recursively.
• We will still use An-1to represent the previous term,
but instead of adding/subtracting r we will be
multiplying by r.
• General form for Recursive Geometric Sequence:
An = An-1 × r
• Example: A = 1,5,25,125,...
n
r=5
Recursive Formula: An = An-1 ×5
EXPLICIT FORMULA
• Explicit Formulas do not require previous terms.
• For geometric sequences, we can write explicit
formulas similarly to how we created exponential
functions.
• Why do you think we can use exponential functions to help
us define geometric sequences?
• The formulas is represented symbolically as:
An = A1 (r)n-1, A1is the first term, r is the growth or decay
factor and n stays n.
• Example: An = 64,16,4,1,...
A1 = 64
r=¼
A = 64(1/ 4)n-1
n
YOU TRY!
An = 3,6,9,12,...
• Is this sequence an arithmetic or geometric
sequence?
• What is the common difference or growth/decay
factor?