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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 Course 3 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.) Course 3 13-3 Other Sequences Learn to find patterns in sequences. Course 3 13-3 Other InsertSequences Lesson Title Here Vocabulary first differences second differences Fibonacci sequence Course 3 13-3 Other Sequences The first five triangular numbers are shown below. 1 Course 3 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 Course 3 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. Course 3 4 4 13-3 Other Sequences Remember! The second difference is the difference between the first differences. Course 3 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. Course 3 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. Course 3 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. Course 3 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. Course 3 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. Course 3 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 Course 3 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. Course 3 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. Course 3 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. Course 3 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. Course 3 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. Course 3 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. Course 3 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. Course 3 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. Course 3 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, Course 3 1, 2, 3, 5, 8, 13, 21, . . . 13-3 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). Course 3 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). Course 3 13-3 Other InsertSequences 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). Course 3