pre-algebra 12-3 other sequences check 12-2 homework

Pre-Algebra

12-3 Other Sequences

Check 12-2 HOMEWORK

Pre-Algebra


Pre-Algebra HOMEWORK

Page 606 #19-28

Pre-Algebra

12-3 Other SequencesStudents 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


Today’s Learning Goal Assignment

Learn to find patterns in sequences.

Pre-Algebra

12-3 Other Sequences12-3 Other Sequences

Pre-Algebra

Warm UpWarm UpProblem of the DayProblem of the DayLesson PresentationLesson Presentation

Pre-Algebra


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

1,048,576

Pre-Algebra


14

Pre-Algebra


Problem of the DayJust 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


Vocabularyfirst differencessecond differencesFibonacci sequence

Pre-Algebra


The first five triangular numbers are shown below.

1 3 6 10 15

Pre-Algebra


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 7Triangular Number 1 3 6 10 15 21 28

765432First differencesSecond differences 11111

Pre-Algebra


Use first and second differences to find the next three terms in the sequence.A. 1, 8, 19, 34, 53, . . .

Additional Example 1A: Using First and Second Differences

The next three terms are 76, 103, 134.

Sequence 1 8 19 34 531st Differences2nd Differences

7 11 15 194 4 4 4

2376

427

103

431

134

Pre-Algebra


Use first and second differences to find the next three terms in the sequence.A. 2, 4, 10, 20, 34, . . .

Try This: Example 1A



2 6 10 144 4 4 4

1852

422

74

426

100

Pre-Algebra


Use first and second differences to find the next three terms in the sequence.B. 12, 15, 21, 32, 50, . . .

Additional Example 1B: Using First and Second Differences



3 6 11 183 5 7 9

2777

1138

115

1351

166

Pre-Algebra


Use first and second differences to find the next three terms in the sequence.B. 2, 2, 3, 6, 12, . . .

Try This: Example 1B



0 1 3 61 2 3 4

1022

515

37

621

58

Pre-Algebra


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


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, . . .

Additional Example 2A: Finding a Rule, Given Terms of a Sequence

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


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, . . .

Try This: Example 2A

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


Give the next three terms in the sequence, using the simplest rule you can find.B. , , , , , . . .

Additional Example 2B: Finding a Rule, Given Terms of a Sequence

25

37

49

511

613

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 + 3 715

817

919The next three terms are , , .

Pre-Algebra


Give the next three terms in the sequence, using the simplest rule you can find.B. 1, 2, 3, 5, 7, 11, . . .

Try This: Example 2B

One possible rule could be the prime numbers from least to greatest.


Pre-Algebra


Give the next three terms in the sequence, using the simplest rule you can find.C. 1, 11, 6, 16, 11, 21, . . .

Additional Example 2C: Finding a Rule, Given Terms of a Sequence

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


Give the next three terms in the sequence, using the simplest rule you can find.C. 101, 1001, 10001, 100001, . . .

Try This: Example 2C

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


Give the next three terms in the sequence, using the simplest rule you can find.D. 1, –2, 3, –4, 5, –6, . . .

Additional Example 2D: Finding a Rule, Given Terms of a Sequence

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


Give the next three terms in the sequence, using the simplest rule you can find.D. 1, 8, 22, 50, 106, . . .

Try This: Example 2D

A rule for this sequence could be to add 3 then multiply by 2.The next three terms are 218, 442, 890.

Pre-Algebra


Find the first five terms of the sequence defined by an = n (n – 2).

Additional Example 3: Finding Terms of a Sequence Given a Rule

a1 = 1(1 – 2) = –1a2 = 2(2 – 2) = 0a3 = 3(3 – 2) = 3a4 = 4(4 – 2) = 8a5 = 5(5 – 2) = 15

The first five terms are –1, 0, 3, 8 , 15.

Pre-Algebra


Find the first five terms of the sequence defined by an = n(n + 2).

Try This: Example 3

a1 = 1(1 + 2) = 3a2 = 2(2 + 2) = 8a3 = 3(3 + 2) = 15a4 = 4(4 + 2) = 24a5 = 5(5 + 2) = 35The first five terms are 3, 8, 15, 24, 35.

Pre-Algebra


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, . . .

Pre-Algebra


Suppose a, b, c, and d are four consecutive numbers in the Fibonacci sequence. Complete the following table and guess the pattern.

Additional Example 4: Using the Fibonacci Sequence

3, 5, 8, 13

13, 21, 34, 55

55, 89, 144, 233

53≈ 1.667 13

8 ≈ 1.6252113 ≈ 1.615 55

34 ≈ 1.618 89 55 ≈ 1.618 233

144 ≈ 1.618

a, b, c, d ba

dc

The ratios are approximately equal to 1.618 (the golden ratio).

Pre-Algebra


Suppose a, b, c, and d are four consecutive numbers in the Fibonacci sequence. Complete the following table and guess the pattern.

Try This: Example 4

4, 7, 11, 18

18, 29, 47, 76

76, 123, 199, 322

74≈ 1.750 18

11 ≈ 1.6362918 ≈ 1.611 76

47 ≈ 1.617123 76 ≈ 1.618 322

199 ≈ 1.618

The ratios are approximately equal to 1.618 (the golden ratio).

a, b, c, d ba

dc

Pre-Algebra


Lesson Quiz1. Use the first and second differences to find the next three terms in the sequence. 2, 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).

37, 50, 65

408, 522, 650

2, 6, 12, 20, 30

pre-algebra 12-3 other sequences check 12-2 homework

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