1. Learn to find terms in an arithmetic sequence.
OBJECTIVES:
Learn to find terms in an arithmetic sequence.
Learn to find terms in a geometric sequence.
Learn to find patterns in sequences.
Learn to represent functions with tables, graphs, or equations.

2. Other Sequences 12-3 Warm Up
Pre-Algebra
12-3
Other Sequences
Warm Up
1. Determine if the sequence could be geometric. If so, give the common ratio: 10, 24, 36, 48, 60, . . .
2. Find the 12th term in the geometric sequence: , 1, 4, 16, . . . no 1 4
1,048,576

3. Problem of the Day
Just by seeing one term, Angela was able to tell whether a certain sequence was geometric or arithmetic. What was the term, and which kind of sequence was it?
0; arithmetic sequence (There is no unique common ratio that would create a geometric sequence.)

4. Vocabulary
first differences
second differences
Fibonacci sequence

5. The first five triangular numbers are shown below.1 3 6 10 15

6. To continue the sequence, you can draw the triangles, or you can look for a pattern. If you subtract every term from the one after it, the first differences create a new sequence. If you do not see a pattern, you can repeat the process and find the second differences.
Term 1 2 3 4 5 6 7
Triangular Number 10 15 21 28
First differences 2 3 4 5 6 7
Second differences 1 1 1 1 1

7. Additional Example 1A: Using First and Second Differences
Use first and second differences to find the next three terms in the sequence.
A. 1, 8, 19, 34, 53, . . .
Sequence 1 8 19 34 53 76
103
134
1st Differences 7 11 15 19 23 27 31
2nd Differences 4 4 4 4 4 4
The next three terms are 76, 103, 134.

8. Sequence 1st Differences 2nd Differences Try This: Example 1A
Use first and second differences to find the next three terms in the sequence.
A. 2, 4, 10, 20, 34, . . .
Sequence 2 4 10 20 34 52 74
100
1st Differences 2 6 10 14 18 22 26
2nd Differences 4 4 4 4 4 4
The next three terms are 52, 74, 100.

9. Additional Example 1B: Using First and Second Differences
Use first and second differences to find the next three terms in the sequence.
B. 12, 15, 21, 32, 50, . . .
Sequence 12 15 21 32 50 77
115
166
1st Differences 3 6 11 18 27 38 51
2nd Differences 3 5 7 9 11 13
The next three terms are 77, 115, 166.

10. Sequence 1st Differences 2nd Differences Try This: Example 1B
Use first and second differences to find the next three terms in the sequence.
B. 2, 2, 3, 6, 12, . . .
Sequence 2 3 6 12 22 37 58
1st Differences 1 3 6 10 15 21
2nd Differences 1 2 3 4 5 6
The next three terms are 22, 37, 58.

11. By looking at the sequence 1, 2, 3, 4, 5,
By looking at the sequence 1, 2, 3, 4, 5, . . ., you would probably assume that the next term is 6. In fact, the next term could be any number. If no rule is given, you should use the simplest recognizable pattern in the given terms.

12. Additional Example 2A: Finding a Rule, Given Terms of a Sequence
Give the next three terms in the sequence, using the simplest rule you can find.
A. 1, 2, 1, 1, 2, 1, 1, 1, 2, . . .
One possible rule is to have one 1 in front of the 1st 2, two 1s in front of the 2nd 2, three 1s in front of the 3rd 2, and so on.
The next three terms are 1, 1, 1.

13. Try This: Example 2A
Give the next three terms in the sequence, using the simplest rule you can find.
A. 1, 2, 3, 2, 3, 4, 3, 4, 5, . . .
One possible rule could be to increase each number by 1 two times then repeat the second to last number.
The next three terms are 4, 5, 6.

14. Additional Example 2B: Finding a Rule, Given Terms of a Sequence
Give the next three terms in the sequence, using the simplest rule you can find.
B. , , , , , . . . 2 5 3 7 4 9 5 11 6 13
One possible rule is to add 1 to the numerator and add 2 to the denominator of the previous term. This could be written as the algebraic rule.
an =
n + 1
2n + 3 7 15 8 17 9 19
The next three terms are , , .

15. Try This: Example 2B
Give the next three terms in the sequence, using the simplest rule you can find.
B. 1, 2, 3, 5, 7, 11, . . .
One possible rule could be the prime numbers from least to greatest.
The next three terms are 13, 17, 19.

16. Additional Example 2C: Finding a Rule, Given Terms of a Sequence
Give the next three terms in the sequence, using the simplest rule you can find.
C. 1, 11, 6, 16, 11, 21, . . .
A rule for the sequence could be to start with 1 and use the pattern of adding 10, subtracting 5 to get the next two terms.
The next three terms are 16, 26, 21.

17. Try This: Example 2C
Give the next three terms in the sequence, using the simplest rule you can find.
C. 101, 1001, 10001, , . . .
A rule for the sequence could be to start and end with 1 beginning with one zero in between, then adding 1 zero to the next number.
The next three terms are , ,

18. Additional Example 2D: Finding a Rule, Given Terms of a Sequence
Give the next three terms in the sequence, using the simplest rule you can find.
D. 1, –2, 3, –4, 5, –6, . . .
A rule for the sequence could be the set of counting numbers with every even number being multiplied by –1.
The next three terms are 7, –8, 9.

19. Try This: Example 2D
Give the next three terms in the sequence, using the simplest rule you can find.
D. 1, 8, 22, 50, 106, . . .
A rule for this sequence could be to add 3 then multiply by 2.
The next three terms are 218, 442, 890.

20. Additional Example 3: Finding Terms of a Sequence Given a Rule
Find the first five terms of the sequence defined by an = n (n – 2).
a1 = 1(1 – 2) = –1
a2 = 2(2 – 2) = 0
a3 = 3(3 – 2) = 3
a4 = 4(4 – 2) = 8
a5 = 5(5 – 2) = 15
The first five terms are –1, 0, 3, 8 , 15.

21. Try This: Example 3
Find the first five terms of the sequence defined by an = n(n + 2).
a1 = 1(1 + 2) = 3
a2 = 2(2 + 2) = 8
a3 = 3(3 + 2) = 15
a4 = 4(4 + 2) = 24
a5 = 5(5 + 2) = 35
The first five terms are 3, 8, 15, 24, 35.

22. A famous sequence called the Fibonacci sequence is defined by the following rule: Add the two previous terms to find the next term.
1, 1, 2, 3, 5, 8, 13, 21, . . .

23. Additional Example 4: Using the Fibonacci Sequence
Suppose a, b, c, and d are four consecutive numbers in the Fibonacci sequence. Complete the following table and guess the pattern. ba dc
a, b, c, d 5 3
≈ 1.667 13 8
≈ 1.625
3, 5, 8, 13 21 13
≈ 1.615 55 34
≈ 1.618
13, 21, 34, 55 89 55
≈ 1.618
233
144
≈ 1.618
55, 89, 144, 233
The ratios are approximately equal to (the golden ratio).

24. Try This: Example 4
Suppose a, b, c, and d are four consecutive numbers in the Fibonacci sequence. Complete the following table and guess the pattern. ba dc
a, b, c, d 7 4
≈ 1.750 18 11
≈ 1.636
4, 7, 11, 18 29 18
≈ 1.611 76 47
≈ 1.617
18, 29, 47, 76
123 76
≈ 1.618
322
199
≈ 1.618
76, 123, 199, 322
The ratios are approximately equal to (the golden ratio).

25. Lesson Quiz
1. Use the first and second differences to find the next three terms in the sequence , 18, 48, 92, 150, 222, 308, . . .
2. Give the next three terms in the sequence, using the simplest rule you can find.
2, 5, 10, 17, 26, . . .
3. Find the first five terms of the sequence defined by an = n(n + 1).
408, 522, 650
37, 50, 65
2, 6, 12, 20, 30