ITERATIVE AND RECURSIVE PATTERNSLesson 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 = -58(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. 1 x 6 + 6 = 3 x 42 x 7 + 6 = 4 x 53 x 8 + 6 = 5 x 64 x

9 + 6 = 6 x 7

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 4- SOLUTIONS The sequence of equations shown below is called a

Tunja sequence. 1 x 6 + 6 = 3 x 42 x 7 + 6 = 4 x 53 x 8 + 6 = 5 x 64 x 9

+ 6 = 6 x 7a. Write the next two equations in the sequence.

5 x 10 + 6 = 7 x 86 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)

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 arithmeticB. 100, 50, 12.5, 1.6, . . .Divide by 2, divide by 4…not arithmeticC. 3, 10, 17, 24, . . .Add 7, add 7, add 7…arithmeticD. –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…arithmeticB. {1, 3, 9, 27, 81, ...}Multiply by 3, multiply by 3…not arithmeticC. {3, -5, 7, -9, 11, ...}Odds, positive, negative…not arithmeticD. {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 + 3D.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 + 3D.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. 896B. 1024C. 1536D. 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. 896B. 1024C. 1536D. 2048

EXAMPLE 10

What is the first term in the sequence below? {___, ___, ___,81, 243, 729, ...}

A. 1B. 3C. 9D. 2

EXAMPLE 10- SOLUTION

What is the first term in the sequence below? {___, ___, ___,81, 243, 729, ...}

A. 1B. 3C. 9D. 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. 1B. 9C. 12D. 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. 1B. 9C. 12D. 22

|2n – 8| = 102n – 8 = 102n = 18n = 9

EXAMPLE 12

Given an + 1= 2, an + 3 and a6 = 3, what is a7?

A. 17B. 12C. 9D. 5

EXAMPLE 12- SOLUTION

Given an + 1= 2, an + 3 and a6 = 3, what is a7?

A. 17B. 12C. 9D. 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, 48B. 38, 50C. 40, 50D. 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, 48B. 38, 50C. 40, 50D. 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 – 4B. n + 5C. 2nD. 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 – 4B. n + 5C. 2nD. 2n – 9

5 10 15 20 N

1 6 11 16 ?