Transcript 12-3

12-3 Other Sequences
Check 12-2 HOMEWORK
Pre-Algebra
12-3 Other Sequences
Pre-Algebra HOMEWORK
Page 606
#19-28
Pre-Algebra
12-3 Other Sequences
Students will be able to solve sequences and represent
functions by completing the following assignments.
•
•
•
•
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.
Pre-Algebra
12-3 Other Sequences
Today’s Learning Goal Assignment
Learn to find
patterns in
sequences.
Pre-Algebra
12-3
12-3Other
OtherSequences
Sequences
Warm Up
Problem of the Day
Lesson Presentation
Pre-Algebra
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, . . . no
2. Find the 12th term in the geometric
sequence: 1 , 1, 4, 16, . . . 1,048,576
4
Pre-Algebra
12-3 Other Sequences
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.)
Pre-Algebra
12-3 Other Sequences
Vocabulary
first differences
second differences
Fibonacci sequence
Pre-Algebra
12-3 Other Sequences
The first five triangular numbers are shown below.
1
3
Pre-Algebra
6
10
15
12-3 Other Sequences
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
Triangular Number
First differences
Second differences
Pre-Algebra
1
2
3
1
3
6
2
3
1
4
6
7
10 15 21 28
4
1
5
5
1
6
1
7
1
12-3 Other Sequences
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
1st Differences
2nd Differences
8
7
19
11
4
34
53
15 19
4
4
76 103 134
23 27 31
4
The next three terms are 76, 103, 134.
Pre-Algebra
4
4
12-3 Other Sequences
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
1st Differences
2nd Differences
4
2
10
6
4
20
10
4
34
14
4
52
18 22 26
4
The next three terms are 52, 74, 100.
Pre-Algebra
74
4
4
100
12-3 Other Sequences
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
1st Differences
2nd Differences
15 21
32
6
11 18
3
3
5
7
50
77 115 166
27 38 51
9
11 13
The next three terms are 77, 115, 166.
Pre-Algebra
12-3 Other Sequences
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
1st Differences
2nd Differences
2
0
3
1
1
6
3
2
12
6
3
22 37
10 15 21
4
The next three terms are 22, 37, 58.
Pre-Algebra
58
5
6
12-3 Other Sequences
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.
Pre-Algebra
12-3 Other Sequences
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.
Pre-Algebra
12-3 Other Sequences
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.
Pre-Algebra
12-3 Other Sequences
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 , 3 , 4 , 5 , 6 , . . .
5 7 9 11 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.
n+1
an = 2n
+3
The next three terms are 7 , 8 , 9 .
15 17 19
Pre-Algebra
12-3 Other Sequences
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.
Pre-Algebra
12-3 Other Sequences
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.
Pre-Algebra
12-3 Other Sequences
Try This: Example 2C
Give the next three terms in the sequence,
using the simplest rule you can find.
C. 101, 1001, 10001, 100001, . . .
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 1000001, 10000001,
100000001.
Pre-Algebra
12-3 Other Sequences
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.
Pre-Algebra
12-3 Other Sequences
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.
Pre-Algebra
12-3 Other Sequences
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.
Pre-Algebra
12-3 Other Sequences
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.
Pre-Algebra
12-3 Other Sequences
A famous sequence called the Fibonacci sequence
is defined by the following rule: Add the two
previous terms to find the next term.
1,
Pre-Algebra
1,
2,
3,
5,
8,
13,
21, . . .
12-3 Other Sequences
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.
a, b, c, d
3, 5, 8, 13
13, 21, 34, 55
55, 89, 144, 233
b
a
5 ≈ 1.667
3
21 ≈ 1.615
13
89 ≈ 1.618
55
d
c
13 ≈ 1.625
8
55 ≈ 1.618
34
233 ≈ 1.618
144
The ratios are approximately equal to 1.618 (the
golden ratio).
Pre-Algebra
12-3 Other Sequences
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.
a, b, c, d
4, 7, 11, 18
18, 29, 47, 76
76, 123, 199, 322
b
a
7 ≈ 1.750
4
29 ≈ 1.611
18
123 ≈ 1.618
76
d
c
18 ≈ 1.636
11
76 ≈ 1.617
47
322 ≈ 1.618
199
The ratios are approximately equal to 1.618 (the
golden ratio).
Pre-Algebra
12-3 Other Sequences
Lesson Quiz
1. Use the first and second differences to find
the next three terms in the sequence.
2, 18, 48, 92, 150, 222, 308, . . .
408, 522, 650
2. Give the next three terms in the sequence,
using the simplest rule you can find.
2, 5, 10, 17, 26, . . .
37, 50, 65
3. Find the first five terms of the sequence
2, 6, 12, 20, 30
defined by an = n(n + 1).
Pre-Algebra