1. ITERATIVE AND RECURSIVE PATTERNS
Lesson 19

2. WARM UP Evaluate each expression [-2|3 + 5|] + [6|3 – 5|]
|3xy + x| for x = -3, y = 8
8x – 4|xy – 6y|

3. WARM UP- SOLUTION Evaluate each expression [-2|3 + 5|] + [6|3 – 5|]
[-2(8) + 6(2)] -4
|3xy + y| for x = -3, y = 8
|3(-3)(8) + 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| |
32 – 4|10|
32 – 40 = -8

4. EXAMPLE 1
Identify the pattern
2, 5, 10, 17

5. EXAMPLE 1- SOLUTION Identify the pattern 2 + 3 = 5 5 + 5 = 10
2, 5, 10, 17
2 + 3 = 5
5 + 5 = 10
= 17
Add 3, add 5, add 7…

6. EXAMPLE 2 Identify each pattern 1, 3, 7, 13, 21… 1, 1, 2, 3, 5, 8, 13…
1, 4, 9, 16, 25, 36…

7. 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, = 3, = 8, = 13
1, 4, 9, 16, 25, 36…
12, 22, 32, 42, 52, 62
Or add the odds

8. 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?

9. 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

10. EXAMPLE 4
The sequence of equations shown below is called a Tunja sequence.
1 x = 3 x 4
2 x = 4 x 5
3 x = 5 x 6
4 x = 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.

11. EXAMPLE 4- SOLUTIONS
The sequence of equations shown below is called a Tunja sequence.
1 x = 3 x 4
2 x = 4 x 5
3 x = 5 x 6
4 x = 6 x 7
Write the next two equations in the sequence.
5 x = 7 x 8
6 x = 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 = 19 x 20
c. Write the equation in the sequence that begins with 100.
100 x = 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)

12. TYPES OF SEQUENCES Arithmetic Geometric
Sequences that are created by adding or subtracting the same number.
Geometric
Sequences that are created by multiplying or dividing the same number.

13. 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 , . . .

14. EXAMPLE 5- SOLUTION Which is an arithmetic sequence?
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

15. 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, ...}

16. 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

17. EXAMPLE 7
Which expression is the nth term of the quadratic sequence shown in the table below?
Term number
Value 1 2 4 3 9 16 5 25 n2
2n2
n2 + 3
2n2 + 2

18. EXAMPLE 7- SOLUTION
Which expression is the nth term of the quadratic sequence shown in the table below?
Term number
Value 1 2 4 3 9 16 5 25 n2
2n2
n2 + 3
2n2 + 2

19. 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

20. 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

21. 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

22. 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

23. EXAMPLE 10
What is the first term in the sequence below? {___, ___, ___,81, 243, 729, ...}
A. 1
B. 3
C. 9
D. 2

24. EXAMPLE 10- SOLUTION
What is the first term in the sequence below? {___, ___, ___,81, 243, 729, ...}
A. 1
B. 3
C. 9
D. 2

25. 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

26. 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

27. EXAMPLE 12 Given an + 1= 2, an + 3 and a6 = 3, what is a7? A. 17 B. 12
C. 9
D. 5

28. EXAMPLE 12- SOLUTION Given an + 1= 2, an + 3 and a6 = 3, what is a7?
B. 12
C. 9
D. 5

29. 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

30. 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

31. 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 ?

32. 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 ?