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Check 12-2 HOMEWORK

In this lesson, students will learn to find terms in arithmetic and geometric sequences, identify patterns in sequences, and represent functions using tables, graphs, or equations.

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Check 12-2 HOMEWORK

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  1. Check 12-2 HOMEWORK

  2. Pre-Algebra HOMEWORK Page 606 #19-28

  3. 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.

  4. Today’s Learning Goal Assignment Learn to find patterns in sequences.

  5. Other Sequences 12-3 Warm Up Problem of the Day Lesson Presentation Pre-Algebra

  6. Other Sequences 12-3 1 4 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

  7. 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.)

  8. Vocabulary first differences second differences Fibonacci sequence

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

  10. 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. First differences 2 3 4 5 6 7 Second differences 1 1 1 1 1

  11. 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, . . . 7 11 15 19 4 4 4 The next three terms are 76, 103, 134.

  12. Try This: Example 1A Use first and second differences to find the next three terms in the sequence. A. 2, 4, 10, 20, 34, . . . 2 6 10 14 4 4 4 The next three terms are 52, 74, 100.

  13. 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, . . . 3 6 11 18 3 5 7 The next three terms are 77, 115, 166.

  14. Try This: Example 1B Use first and second differences to find the next three terms in the sequence. B. 2, 2, 3, 6, 12, . . . 0 1 3 6 1 2 3 The next three terms are 22, 37, 58.

  15. 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.

  16. 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.

  17. 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.

  18. 8 9 2 7 3 4 5 6 13 15 5 7 9 11 19 17 an = n + 1 2n + 3 The next three terms are , , . 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. , , , , , . . . 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.

  19. 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.

  20. 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.

  21. 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.

  22. 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.

  23. 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.

  24. 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.

  25. 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.

  26. 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, . . .

  27. 233 89 55 5 21 13 55 34 8 13 3 144 ≈ 1.625 ≈ 1.667 ≈ 1.618 ≈ 1.618 ≈ 1.615 ≈ 1.618 dc ba 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. 3, 5, 8, 13 13, 21, 34, 55 55, 89, 144, 233 The ratios are approximately equal to 1.618 (the golden ratio).

  28. 18 76 29 7 123 322 4 11 199 76 47 18 ≈ 1.618 ≈ 1.618 ≈ 1.611 ≈ 1.750 ≈ 1.617 ≈ 1.636 dc ba 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. 4, 7, 11, 18 18, 29, 47, 76 76, 123, 199, 322 The ratios are approximately equal to 1.618 (the golden ratio).

  29. 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, . . . 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

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