Transcript 13-3 Other Sequences

13-3 Other Sequences
Warm Up
Problem of the Day
Lesson Presentation
Course 3
13-3 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
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13-3 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.)
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13-3 Other Sequences
Learn to find patterns in sequences.
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13-3 Other
InsertSequences
Lesson Title Here
Vocabulary
first differences
second differences
Fibonacci sequence
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13-3 Other Sequences
The first five triangular numbers are shown below.
1
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3
6
10
15
13-3 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
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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
13-3 Other Sequences
Additional Example 1A: Using First and Second
Differences
Use first and second differences to find the
next three terms in the sequence.
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.
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4
4
13-3 Other Sequences
Remember!
The second difference is the difference between
the first differences.
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13-3 Other Sequences
Additional Example 1B: Using First and Second
Differences
Use first and second differences to find the
next three terms in the sequence.
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.
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13-3 Other Sequences
Check It Out: Example 1A
Use first and second differences to find the
next three terms in the sequence.
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.
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74
4
4
100
13-3 Other Sequences
Check It Out: Example 1B
Use first and second differences to find the
next three terms in the sequence.
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.
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58
5
6
13-3 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.
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13-3 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.
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.
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13-3 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
2, 3, 4, 5 , 6 , . . .
5 7 9 11 13
Add 1 to the numerator and add 2 to the
denominator of the previous term. This could be
n+1
a
=
written as the algebraic rule n 2n + 3 .
The next three terms are 7 , 8 , 9 .
15 17 19
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13-3 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.
1, 11, 6, 16, 11, 21, . . .
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.
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13-3 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.
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.
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13-3 Other Sequences
Check It Out: Example 2A
Give the next three terms in the sequence,
using the simplest rule you can find.
1, 2, 3, 2, 3, 4, 3, 4, 5, . . .
Increase each number by 1 two times then
repeat the second to last number.
The next three terms are 4, 5, 6.
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13-3 Other Sequences
Check It Out: Example 2B
Give the next three terms in the sequence,
using the simplest rule you can find.
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.
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13-3 Other Sequences
Check It Out: Example 2C
Give the next three terms in the sequence,
using the simplest rule you can find.
101, 1001, 10001, 100001, . . .
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.
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13-3 Other Sequences
Check It Out: Example 2D
Give the next three terms in the sequence,
using the simplest rule you can find.
1, 8, 22, 50, 106, . . .
Add 3 to the previous term and then multiply
by 2. This could be written as the algebraic
rule an = (3 + an – 1)2.
The next three terms are 218, 442, 890.
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13-3 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.
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13-3 Other Sequences
Check It Out: 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.
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13-3 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,
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1,
2,
3,
5,
8,
13,
21, . . .
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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).
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13-3 Other Sequences
Check It Out: 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).
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Lesson Title Here
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).
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