7.5 Use Recursive Rules with Sequences and Functions

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7.5 Use Recursive Rules with Sequences and
Functions

• p. 467

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

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Explicit Rule

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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 a01 and anan-1-2
• The 1st five terms of this sequence would be a0,
  a1, a2, a3, a4 OR
• 1, -1, -3, -5, -7

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Recursive Equations

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Example Write the indicated rule for the
arithmetic sequence with a115 and d5.

• Recursive rule
• (Use the idea that you get the next term by
  adding 5 to the previous term.)
• Or anan-15
• So, a recursive rule would be a115, anan-15

• Explicit rule
• ana1(n-1)d
• an15(n-1)5
• an155n-5
• an105n

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Example Write the indicated rule for the
geometric sequence with a14 and r0.2.

• Explicit rule
• ana1rn-1
• an4(0.2)n-1

• Recursive rule
• (Use the idea that you get the next term by
  multiplying the previous term by 0.2)
• Or anran-10.2an-1
• So, a recursive rule for the sequence would be
  a14, an0.2an-1

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Example Write the 1st 5 terms of the sequence.

• a12, a22, anan-2-an-1
• a3a3-2-a3-1?a1-a22-20
• a4a4-2-a4-1?a2-a32-02
• a5a5-2-a5-1?a3-a40-2-2
• 2, 2, 0, 2, -2

2nd term
1st term
1 2 3 4 5
2 2 0 2 -2
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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
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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
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.
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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
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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
Write the first five terms of the sequence.
4. a1 4, an 2an 1 1
SOLUTION
a1 4
15
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
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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
,
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Write a recursive rule for the sequence.
b. 1, 1, 2, 6, 24, . . .
SOLUTION
This sequence lists factorial numbers.
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Iterating Functions
 
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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 (5)
f (2)
f (16)
3(25) 1
3(16) 1
3(2) 1
5
47
16
20
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
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7.5 Assignment

• p. 470, 3-27 odd, skip 21

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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-1an-2
• So, a recursive rule would be
• a11, a22, an an-1an-2

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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-1an-2)
• So the recursive rule would be
• a11, a21, an 2(an-1an-2)