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ITERATIVE AND
RECURSIVE PATTERNS
Lesson 19
WARM UP

Evaluate each expression

[-2|3 + 5|] + [6|3 – 5|]

|3xy + x| for x = -3, y = 8

8x – 4|xy – 6y|
WARM UP- SOLUTION

Evaluate each expression

[-2|3 + 5|] + [6|3 – 5|]
[-2(8) + 6(2)]
-16 + 12
-4

|3xy + y| for x = -3, y = 8
|3(-3)(8) + 8|
|-72 + 8|
|-64|
64

8x – 4|xy – 6y| for x = 4, y = -5
8(4) – 4|(4)(-5) – 6(-5)|
32 – 4|-20 – -30|
32 – 4|-20 + 30|
32 – 4|10|
32 – 40 = -8
EXAMPLE 1

Identify the pattern

2, 5, 10, 17
EXAMPLE 1- SOLUTION

Identify the pattern

2, 5, 10, 17
2+3=5
 5 + 5 = 10
 10 + 7 = 17


Add 3, add 5, add 7…
EXAMPLE 2

Identify each pattern

1, 3, 7, 13, 21…

1, 1, 2, 3, 5, 8, 13…

1, 4, 9, 16, 25, 36…
EXAMPLE 2- SOLUTIONS

Identify each pattern
1, 3, 7, 13, 21…
 Add 2, add 4, add 6, add 8

1, 1, 2, 3, 5, 8, 13…
 Add the 2 previous numbers to get the next.
 1 + 1 = 2, 1 + 2 = 3, 3 + 5 = 8, 5 + 8 = 13

1, 4, 9, 16, 25, 36…
 12, 22, 32, 42, 52, 62
 Or add the odds

EXAMPLE 3

The numbers in the sequence 2, 7, 12, 17, 22, . . .
increase by fives. The numbers in the sequence 3,
10, 17, 24, 31, . . . increase by sevens. The
number 17 occurs in both sequences.
If the two sequences are continued, what is the
next number that will be seen in both sequences?
EXAMPLE 3- SOLUTION

The numbers in the sequence 2, 7, 12, 17, 22, . . .
increase by fives. The numbers in the sequence 3,
10, 17, 24, 31, . . . increase by sevens. The
number 17 occurs in both sequences.
If the two sequences are continued, what is the
next number that will be seen in both sequences?
2, 7, 12, 17, 22, 27, 32, 37, 42, 47, 52
3, 10, 17, 24, 31, 38, 45, 52
EXAMPLE 4

The sequence of equations shown below is called a
Tunja sequence.

1x6+6=3x4 2x7+6=4x5 3x8+6=5x6 4x9+6
=6x7
a. Write the next two equations in the sequence.
b. The first four equations in the sequence begin with 1,
2, 3, and 4. Write the equation in the sequence that
begins with 17.
c. Write the equation in the sequence that begins with
100.
d. Write the equation in the sequence that begins with
n. Show or explain how you obtained your answer.
EXAMPLE
4SOLUTIONS
 The sequence of equations shown below is called a Tunja
sequence.

1x6+6=3x4 2x7+6=4x5 3x8+6=5x6 4x9+6=6
x7
Write the next two equations in the sequence.
5 x 10 + 6 = 7 x 8
6 x 11 + 6 = 8 x 9
b. The first four equations in the sequence begin with 1, 2, 3,
and 4. Write the equation in the sequence that begins with
17.
17 x 22 + 6 = 19 x 20
c. Write the equation in the sequence that begins with 100.
100 x 105 + 6 = 102 x 103
d. Write the equation in the sequence that begins with n.
Show or explain how you obtained your answer.
n x (n + 5) + 6 = (n + 2) x (n + 3)
a.
TYPES OF SEQUENCES

Arithmetic


Sequences that are created by adding or subtracting
the same number.
Geometric

Sequences that are created by multiplying or dividing
the same number.
EXAMPLE 5

Which is an arithmetic sequence?
A. 2, 5, 9, 14, . . .
B. 100, 50, 12.5, 1.6, . . .
C. 3, 10, 17, 24, . . .
D. –2, –1, –1/2 , –1/4 , . . .
EXAMPLE 5- SOLUTION

Which is an arithmetic sequence?
A. 2, 5, 9, 14, . . .
Add 3, add 4, add 5…not arithmetic
B. 100, 50, 12.5, 1.6, . . .
Divide by 2, divide by 4…not arithmetic
C. 3, 10, 17, 24, . . .
Add 7, add 7, add 7…arithmetic
D. –2, –1, –1/2 , –1/4 , . . .
Divide by 2, divide by 2, divide by 2…not
arithmetic
EXAMPLE 6

Which of the following sets represents an
arithmetic sequence?
A. {2, 11, 20, 29, 38, ...}
B. {1, 3, 9, 27, 81, ...}
C. {3, -5, 7, -9, 11, ...}
D. {1, 16, 36, 64, 100, ...}
EXAMPLE 6- SOLUTION

Which of the following sets represents an
arithmetic sequence?
A. {2, 11, 20, 29, 38, ...}
Add 9, add 9, add 9…arithmetic
B. {1, 3, 9, 27, 81, ...}
Multiply by 3, multiply by 3…not arithmetic
C. {3, -5, 7, -9, 11, ...}
Odds, positive, negative…not arithmetic
D. {1, 16, 36, 64, 100, ...}
Perfect squares…not arithmetic
EXAMPLE 7

Which expression is the nth term of the quadratic
sequence shown in the table below?
Term
number
Value
1
1
2
4
3
9
4
16
5
25
A. n2
B. 2n2
C. n2 + 3
D.2n2 + 2
EXAMPLE 7- SOLUTION

Which expression is the nth term of the quadratic
sequence shown in the table below?
Term
number
Value
1
1
2
4
3
9
4
16
5
25
A. n2
B. 2n2
C. n2 + 3
D.2n2 + 2
EXAMPLE 8

Sandra wrote the sequence below. 2, 5, 10, 17, . .
. Which equation represents the rule for finding
the nth term of this sequence?
A. an = n+1
B. an = 2n2
C. an = n2 + 1
D. an = 2n + 1
EXAMPLE 8- SOLUTION

Sandra wrote the sequence below. 2, 5, 10, 17, . .
. Which equation represents the rule for finding
the nth term of this sequence?
A. an = n+1
B. an = 2n2
C. an = n2 + 1
D. an = 2n + 1
EXAMPLE 9
The first five terms in a geometric sequence are
shown below.
2, 8, 32, 128, 512, . . .
What is the next term in the sequence?

A. 896
B. 1024
C. 1536
D. 2048
EXAMPLE 9- SOLUTION
The first five terms in a geometric sequence are
shown below.
2, 8, 32, 128, 512, . . .
What is the next term in the sequence?

A. 896
B. 1024
C. 1536
D. 2048
EXAMPLE 10

What is the first term in the sequence
below? {___, ___, ___,81, 243, 729, ...}
A. 1
B. 3
C. 9
D. 2
EXAMPLE 10- SOLUTION

What is the first term in the sequence
below? {___, ___, ___,81, 243, 729, ...}
A. 1
B. 3
C. 9
D. 2
EXAMPLE 11
The sequence below uses the rule an = |2n – 8|,
beginning with a1.
{6, 4, 2, 0, 2, 4, ...}
If an = 10, what is the value of n?

A. 1
B. 9
C. 12
D. 22
EXAMPLE 11- SOLUTION
The sequence below uses the rule an = |2n – 8|,
beginning with a1.
{6, 4, 2, 0, 2, 4, ...}
If an = 10, what is the value of n?

A. 1
B. 9
C. 12
D. 22
|2n – 8| = 10
2n – 8 = 10
2n = 18
n=9
EXAMPLE 12
Given
an + 1= 2, an + 3 and a6 = 3, what is a7?

A. 17
B. 12
C. 9
D. 5
EXAMPLE 12- SOLUTION
Given
an + 1= 2, an + 3 and a6 = 3, what is a7?

A. 17
B. 12
C. 9
D. 5
EXAMPLE 13
Jen wrote the pattern shown below.
10, 12, 16, 22, ...
If the pattern continues, what will be the 6th and
7th terms of the original pattern?

A. 38, 48
B. 38, 50
C. 40, 50
D. 40, 52
EXAMPLE 13- SOLUTION
Jen wrote the pattern shown below.
10, 12, 16, 22, ...
If the pattern continues, what will be the 6th and
7th terms of the original pattern?

A. 38, 48
B. 38, 50
C. 40, 50
D. 40, 52
10, 12, 16, 22, 30, 40, 52
Add 2, 4, 6, 8, 10, 12
EXAMPLE 14

The nth term of the linear pattern defined by the
table is given by which equation?
A. n – 4
B. n + 5
C. 2n
D. 2n – 9
5
10
15
20
N
1
6
11
16
?
EXAMPLE 14- SOLUTION

The nth term of the linear pattern defined by the
table is given by which equation?
A. n – 4
B. n + 5
C. 2n
D. 2n – 9
5
10
15
20
N
1
6
11
16
?