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 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

Recursive Equations  

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. 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 Explicit rule an=a1+(n-1)d an=15+(n-1)5 an=15+5n-5 an=10+5n

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. 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 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

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 a4 = a3 + 4 = 13 + 4 = 17 a5 = 3a4 = 3(27) = 81 a5 = a4 + 4 = 17 + 4 = 21 a6 = 3a5 = 3(81) = 243

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. ANSWER So, a recursive rule for the sequence is a1 = 3, an = an – 1 + 10.

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

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

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 ANSWER 1, 2, 4, 7, 11

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

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 r = a2 a1 = 7 an = r an – 1 = 7 · an – 1 So, a recursive rule for the sequence is ANSWER a1 = 2, an = 7an – 1

Write a recursive rule for the sequence. SOLUTION Beginning with the third term in the sequence, each term is the sum of the two previous terms. a. ANSWER So, a recursive rule is a1 = 1, a2 = 1, an = an – 2 + an – 1. 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,…

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

Iterating Functions  

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 x1 = f (x0) x2 = f (x1) x3 = f (x2) = f (2) = f (–5) = f (16) = –3(2) + 1 = –3(25) + 1 = –3(16) + 1 = – 5 = 16 = – 47 The first three iterates are – 5, 16, and – 47. ANSWER

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) = f (2) = 4 (5) – 3 = 4 (17) – 3 = 8 – 3 = 17 = 68 – 3 = 5 = 65 The first three iterates are 5, 17, and 65. ANSWER

   

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)