1. 13-3 Other Sequences Course 3 Warm Up Warm Up Problem of the Day Problem of the Day Lesson Presentation Lesson Presentation

2. Warm Up 1. Determine if the sequence could be geometric. If so, give the common ratio: 10, 24, 36, 48, 60,... 2. Find the 12 th term in the geometric sequence:, 1, 4, 16,... no 1,048,576 Course 3 13-3 Other Sequences 1 4

3. 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

4. Learn to find patterns in sequences. Course 3 13-3 Other Sequences

5. Vocabulary first differences second differences Fibonacci sequence Insert Lesson Title Here Course 3 13-3 Other Sequences

6. The first five triangular numbers are shown below. 1361015 Course 3 13-3 Other Sequences

7. 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 1234567 Triangular Number 13610152128 765432First differences Second differences 11111 Course 3 13-3 Other Sequences

8. Use first and second differences to find the next three terms in the sequence. 1, 8, 19, 34, 53,... Additional Example 1A: Using First and Second Differences The next three terms are 76, 103, 134. Sequence 18193453 1st Differences 2nd Differences 7 11 1519 4 4 4 4 23 76 4 27 103 4 31 134 Course 3 13-3 Other Sequences

9. Course 3 13-3 Other Sequences The second difference is the difference between the first differences. Remember!

10. Use first and second differences to find the next three terms in the sequence. 12, 15, 21, 32, 50,... Additional Example 1B: Using First and Second Differences The next three terms are 77, 115, 166. Sequence 1215213250 1st Differences 2nd Differences 3 6 1118 3 5 7 9 27 77 11 38 115 13 51 166 Course 3 13-3 Other Sequences

11. Use first and second differences to find the next three terms in the sequence. 2, 4, 10, 20, 34,... Check It Out: Example 1A The next three terms are 52, 74, 100. Sequence 24102034 1st Differences 2nd Differences 2 6 1014 4 4 4 4 18 52 4 22 74 4 26 100 Course 3 13-3 Other Sequences

12. Use first and second differences to find the next three terms in the sequence. 2, 2, 3, 6, 12,... Check It Out: Example 1B The next three terms are 22, 37, 58. Sequence 223612 1st Differences 2nd Differences 0 1 36 1 2 3 4 10 22 5 15 37 6 21 58 Course 3 13-3 Other Sequences

13. 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

14. Give the next three terms in the sequence, using the simplest rule you can find. 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. Course 3 13-3 Other Sequences

15. Give the next three terms in the sequence, using the simplest rule you can find,,,,,... Additional Example 2B: Finding a Rule, Given Terms of a Sequence 2 5 3 7 4 9 5 11 6 13 Add 1 to the numerator and add 2 to the denominator of the previous term. This could be written as the algebraic rule. a n = n + 1 2n + 3 7 15 8 17 9 19 The next three terms are,,. Course 3 13-3 Other Sequences

16. Give the next three terms in the sequence, using the simplest rule you can find. 1, 11, 6, 16, 11, 21,... Additional Example 2C: Finding a Rule, Given Terms of a Sequence 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

17. Give the next three terms in the sequence, using the simplest rule you can find. 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. Course 3 13-3 Other Sequences

18. Give the next three terms in the sequence, using the simplest rule you can find. 1, 2, 3, 2, 3, 4, 3, 4, 5,... Check It Out: Example 2A 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

19. Give the next three terms in the sequence, using the simplest rule you can find. 1, 2, 3, 5, 7, 11,... Check It Out: Example 2B 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

20. Give the next three terms in the sequence, using the simplest rule you can find. 101, 1001, 10001, 100001,... Check It Out: Example 2C 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

21. Give the next three terms in the sequence, using the simplest rule you can find. 1, 8, 22, 50, 106,... Check It Out: Example 2D Add 3 to the previous term and then multiply by 2. This could be written as the algebraic rule a n = (3 + a n – 1 )2. The next three terms are 218, 442, 890. Course 3 13-3 Other Sequences

22. Find the first five terms of the sequence defined by a n = n(n – 2). Additional Example 3: Finding Terms of a Sequence Given a Rule a 1 = 1(1 – 2) = –1 a 2 = 2(2 – 2) = 0 a 3 = 3(3 – 2) = 3 a 4 = 4(4 – 2) = 8 a 5 = 5(5 – 2) = 15 The first five terms are –1, 0, 3, 8, 15. Course 3 13-3 Other Sequences

23. Find the first five terms of the sequence defined by a n = n(n + 2). Check It Out: Example 3 a 1 = 1(1 + 2) = 3 a 2 = 2(2 + 2) = 8 a 3 = 3(3 + 2) = 15 a 4 = 4(4 + 2) = 24 a 5 = 5(5 + 2) = 35 The first five terms are 3, 8, 15, 24, 35. Course 3 13-3 Other Sequences

24. 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,... Course 3 13-3 Other Sequences

25. 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 5 3 ≈ 1.667 13 8 ≈ 1.625 21 13 ≈ 1.615 55 34 ≈ 1.618 89 55 ≈ 1.618 233 144 ≈ 1.618 a, b, c, d baba dcdc The ratios are approximately equal to 1.618 (the golden ratio). Course 3 13-3 Other Sequences

26. Suppose a, b, c, and d are four consecutive numbers in the Fibonacci sequence. Complete the following table and guess the pattern. Check It Out: Example 4 4, 7, 11, 18 18, 29, 47, 76 76, 123, 199, 322 7 4 ≈ 1.750 18 11 ≈ 1.636 29 18 ≈ 1.611 76 47 ≈ 1.617 123 76 ≈ 1.618 322 199 ≈ 1.618 The ratios are approximately equal to 1.618 (the golden ratio). a, b, c, d baba dcdc Course 3 13-3 Other Sequences

27. 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 a n = n(n + 1). 37, 50, 65 408, 522, 650 Insert Lesson Title Here 2, 6, 12, 20, 30 Course 3 13-3 Other Sequences