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7 .5 Use Recursive Rules with Sequences and Functions. p. 467. What is a recursive rule for arithmetic sequences? What is a recursive rule for geometric sequences? What is an iteration?. Explicit Rule. Recursive Rule.

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## 7 .5 Use Recursive Rules with Sequences and Functions

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**What is a recursive rule for arithmetic sequences?**• What is a recursive rule for geometric sequences? • What is an iteration?**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**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.**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.**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**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**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.**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.**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…**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**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**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**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,…**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.**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**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**7.5 Assignment:**p. 470, 3-27 odd, skip 21**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**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)

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