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7 .5 Use Recursive Rules with Sequences and Functions

7 .5 Use Recursive Rules with Sequences and Functions

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7 .5 Use Recursive Rules with Sequences and Functions

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  1. 7.5 Use Recursive Rules with Sequences and Functions p. 467

  2. What is a recursive rule for arithmetic sequences? • What is a recursive rule for geometric sequences? • What is an iteration?

  3. Explicit Rule

  4. Recursive Rule • Gives the beginning term(s) of a sequence and a recursive rule that relates the given term(s) to the next terms in the sequence. • For example: Given a0=1 and an=an-1-2 • The 1st five terms of this sequence would be: a0, a1, a2, a3, a4 OR • 1, -1, -3, -5, -7

  5. Recursive Equations

  6. Explicit rule an=a1+(n-1)d an=15+(n-1)5 an=15+5n-5 an=10+5n Recursive rule (*Use the idea that you get the next term by adding 5 to the previous term.) Or an=an-1+5 So, a recursive rule would be a1=15, an=an-1+5 Example: Write the indicated rule for the arithmetic sequence with a1=15 and d=5.

  7. Explicit rule an=a1rn-1 an=4(0.2)n-1 Recursive rule (*Use the idea that you get the next term by multiplying the previous term by 0.2) Or an=r*an-1=0.2an-1 So, a recursive rule for the sequence would be a1=4, an=0.2an-1 Example: Write the indicated rule for the geometric sequence with a1=4 and r=0.2.

  8. Example: Write the 1st 5 terms of the sequence. • a1=2, a2=2, an=an-2-an-1 a3=a3-2-a3-1→a1-a2=2-2=0 a4=a4-2-a4-1→a2-a3=2-0=2 a5=a5-2-a5-1→a3-a4=0-2=-2 2, 2, 0, 2, -2 2nd term 1st term 1 2 3 4 5 2 2 0 2 -2

  9. Write the first six terms of the sequence. a. a0 = 1, an= an – 1 + 4 b. a1 = 1, an= 3an – 1 SOLUTION a. a0 = 1 b. a1 = 1 a1 = a0 + 4 = 1 + 4 = 5 a2 = 3a1 = 3(1) = 3 a2 = a1 + 4 = 5 + 4 = 9 a3 = 3a2 = 3(3) = 9 a3 = a2 + 4 = 9 + 4 = 13 a4 = 3a3 = 3(9) = 27 a5 = 3a4 = 3(27) = 81 a4 = a3 + 4 = 13 + 4 = 17 a5 = a4 + 4 = 17 + 4 = 21 a6 = 3a5 = 3(81) = 243

  10. ANSWER So, a recursive rule for the sequence isa1 = 3, an= an– 1 + 10. Write the first six terms of the sequence. a. 3, 13, 23, 33, 43, . . . SOLUTION The sequence is arithmetic with first term a1 = 3 and common difference d = 13 – 3 = 10. an= an – 1 + d General recursive equation for an = an – 1 + 10 Substitute 10 for d.

  11. b. The sequence is geometric with first term a1 = 16 and common ratio r = = 2.5. an= ran– 1 40 16 ANSWER So, a recursive rule for the sequence is a1 = 16,an= 2.5an – 1. Write the first six terms of the sequence. b. 16, 40, 100, 250, 625, . . . General recursive equation for an = 2.5an – 1 Substitute 2.5 for r.

  12. 1. a1 = 3, an= an – 1 7 – – – – a2 = a1 7 = 3 7 = 4 ANSWER 3, –4, –11, –18, –25 Write the first five terms of the sequence. SOLUTION a1 = 3 3 −4 −11 −18 −25 a3 = a2– 7 a3 = – 4 – 7 = – 11 a4 = a3– 7 = – 11 – 7 = – 18 a5 = a4– 7 = – 18 – 7 = – 25 Or think of it this way…

  13. ANSWER 1, 2, 4, 7, 11 Write the first five terms of the sequence. 3. a0 = 1, an= an – 1 + n SOLUTION a0 = 1 a1 = a0+ 1 = 1 + 1 = 2 a2 = a1+ 1 = 2+ 2 = 4 a3 = a2+ 3 = 4 + 3 = 7 a4 = a3+ 4 = 7 + 4 = 11

  14. a2 = 2a1– 1 = (2 4) – 1 = 8 – 1 = 7 a3 = 2a2– 1 = (2 7) – 1 = 14 – 1 = 13 a4 = 2a3– 1 = (2 13) – 1 = 26 – 1 = 25 4, 7, 13 25, 49 ANSWER a5 = 2a4– 1 = (2 25) – 1 = 49 Write the first five terms of the sequence. 4. a1 = 4, an= 2an – 1– 1 SOLUTION a1 = 4

  15. an= ran– 1 a2 r = = 7 a1 ANSWER So, a recursive rule for the sequence is a1 = 2, an= 7an – 1 Write a recursive rule for the sequence. 5. 2, 14, 98, 686, 4802, . . . SOLUTION The sequence is geometric with first term a1 = 2 and common ratio = 7 ·an – 1

  16. a. Beginning with the third term in the sequence, each term is the sum of the two previous terms. ANSWER So, a recursive rule is a1 = 1, a2 = 1, an= an – 2 + an–1. Write a recursive rule for the sequence. a. 1, 1, 2, 3, 5, . . . SOLUTION This sequence is the Fibonacci sequence. By definition, the first two numbers in the Fibonacci sequence are 0 and 1 (alternatively, 1 and 1), and each subsequent number is the sum of the previous two. 0,1,1,2,3,5,8,13,21,34,55,89,144,…

  17. b. Denote the first term by a0 = 1. Then note that a1= 1= 1a0, a2= 2 = 2a1, a3= 6 = 3a2, and so on. ANSWER So, a recursive rule isa0 = 1, an= n an – 1. Write a recursive rule for the sequence. b. 1, 1, 2, 6, 24, . . . SOLUTION This sequence lists factorial numbers.

  18. Iterating Functions

  19. ANSWER The first three iterates are– 5, 16, and– 47. Find the first three iterates x1, x2, and x3 of the function f (x) = –3x + 1 for an initial value of x0 = 2. SOLUTION x2 = f (x1) x1 = f (x0) x3 = f (x2) = f (–5) = f (2) = f (16) = –3(25) + 1 = –3(16) + 1 = –3(2) + 1 = – 5 = – 47 = 16

  20. ANSWER The first three iterates are5, 17, and65. Find the first three iterates of the function for the initial value. 11.f (x) = 4x – 3, x0 =2 SOLUTION x1 = f (x0) x2 = f (x1) x3 = f (x2) = 4 (5) – 3 = f (2) = 4(17) – 3 = 68 – 3 = 8 – 3 = 17 =5 = 65

  21. 7.5 Assignment: p. 470, 3-27 odd, skip 21

  22. Write a recursive rule for the sequence 1,2,2,4,8,32,… . • First, notice the sequence is neither arithmetic nor geometric. • So, try to find the pattern. • Notice each term is the product of the previous 2 terms. • Or, an-1*an-2 • So, a recursive rule would be: a1=1, a2=2, an= an-1*an-2

  23. Example: Write a recursive rule for the sequence 1,1,4,10,28,76. • Is the sequence arithmetic, geometric, or neither? • Find the pattern. • 2 times the sum of the previous 2 terms • Or 2(an-1+an-2) • So the recursive rule would be: a1=1, a2=1, an= 2(an-1+an-2)