Download Patterns and Sequences

Warm – Up:
1. Solve for y:
6(x + 4) = 2(y + 5)
2. Gary makes $10 per hour raking leaves plus a flat fee of $20
for the bags that will hold the leaves. Write a function to
represent this scenario. How much money would Gary make if
he work a total of 3 hours?
Patterns and Sequences
Patterns refer to usual types of procedures or rules that can be
followed.
Patterns are useful to predict what came before or what might
come after a set a numbers that are arranged in a particular order.
This arrangement of numbers is called a sequence.
For example:
3,6,9,12 and 15 are numbers that form a pattern called a sequence
The numbers that are in the sequence are called terms.
Patterns and Sequences
Arithmetic sequence– An arithmetic
sequence is a set of numbers put into a
specific order by a pattern of addition or
subtraction.
Arithmetic Sequence
Find the next three numbers or terms in each pattern.
a. 7, 12, 17, 22,...
a. 7, 12, 17, 22,...
5
5
5
Look for a pattern: usually a procedure or rule that uses the
same number or expression each time to find the next term.
The pattern is to add 5 to each term.
The next three terms are:
22  5  27
27  5  32
32  5  37
Arithmetic Sequence
Find the next three numbers or terms in each pattern.
b. 45, 42, 39, 36,...
b. 45, 42, 39, 36,...
 (3)
 (3)
 (3)
Look for a pattern: usually a procedure or rule that uses the
same number or expression each time to find the next term.
The pattern is to add the integer (-3) to each term.
The next three terms are:
36  (3)  33
33  (3)  30
30  (3)  27
Recursive Formula for Arithmetic
Recursive formula is a formula that is used to
determine the next term of a sequence using one or
more of the preceding terms.
The Sequence 2, 5, 8, 11, 14 ….
Recursive
Formula
Write the Recursive Formula for
Arithmetic sequences
• Example 1
5, 8, 11, 14…
• Example 2
32, 27, 22, 17 …
Explicit Formula
• An explicit formula expresses the nth term
of a sequence in terms of n. It can be used
to find any term in the sequence based on
the term number.
Explicit formula for Arithmetic
Write an Explicit formula for
Arithmetic
Write an Explicit formula for
Arithmetic
• Example 1
12, 14, 16, 18
• Example 2
110, 103, 96, 89, 82, 75 ….
Finding the nth term in the
Explicit formula for Arithmetic
Geometric Sequences
Geometric sequence – A sequence of
numbers in which each term is formed by
multiplying the previous term by the same
number or expression.
Geometric Sequence
Find the next three numbers or terms in each pattern.
b. 3, 9, 27, 81,...
b. 3, 9, 27, 81,...
3
3
3
Look for a pattern: usually a procedure or rule that uses the
same number or expression each time to find the next term.
The pattern is to multiply 3 to each term.
The next three terms are:
Geometric Sequence
Find the next three numbers or terms in each pattern.
b. 528, 264, 132, 66... b. 528, 264, 132, 66,...
Look for a pattern: usually a
procedure or rule that uses the same
number or expression each time to
find the next term. The pattern is to
divide by 2 to each term.
The next three terms are:
1
 2 or 
2
 2 or 
1
1
2  2 or  2
Note: To divide by a number is the same
as multiplying by its reciprocal. The
pattern for a geometric sequence is
represented as a multiplication pattern.
For example: to divide by 2 is
represented as the pattern multiply by ½.
… 33, 16.5, 8.25…
Recursive Formula Geometric
The recursive formula for geometric sequences is different
than the recursive formula for arithmetic sequences
The Sequence 5, 25, 125, 625….
Recursive
Formula
Write the Recursive Formula for
Geometric sequences
• Example 1:
297, 99, 33, 11….
• Example 2:
5, 20, 80, 320….
Explicit Formula for Geometric
Write an Explicit formula for
Geometric
Write the Explicit formula for
Geometric
• Example 1:
5, 15, 45, 135…
• Example 2:
1, -6, 36, -216…
Finding the nth term in the
Explicit formula for Geometric