Recursive Functions, Iterates, and Finite Differences

1
Recursive Functions,Iterates, and Finite
Differences
By Jeffrey Bivin Lake Zurich High
School jeff.bivin_at_lz95.org
Last Updated May 21, 2008
2
Recursive Function

• A recursive function is a function whose domain
  is the set of nonnegative integers and is made up
  of two parts
• 1. Start
• 2. Definition

Jeff Bivin -- LZHS
3
Example 1

• a1 5
• an an-1 10

start
definition
n 2 a2 a(2-1) 10 a2 a1 10 a2 5
10 a2 15
n 3 a3 a (3-1) 10 a3 a2 10 a3 15
10 a3 25
n 4 a4 a(4-1) 10 a4 a3 10 a4 25
10 a4 35
Jeff Bivin -- LZHS
4
Example 2

• f(1) 3
• f(n) 5f(n-1) 2

start
definition
n 2 f(2) 5f(2-1) 2 f(2) 5f(1) 2 f(2)
53 2 f(2) 17
n 3 f(3) 5f(3-1) 2 f(3) 5f(2) 2 f(3)
517 2 f(3) 87
n 4 f(4) 5f(4-1) 2 f(4) 5f(3) 2 f(4)
587 2 f(4) 437
Jeff Bivin -- LZHS
5
Example 3

• f(1) 1
• f(2) 1
• f(n) f(n-1) f(n-2)
• f(3) f(3-1) f(3-2) f(2) f(1) 1 1 2
• f(4) f(4-1) f(4-2) f(3) f(2) 2 1 3
• f(5) f(5-1) f(5-2) f(4) f(3) 3 2 5
• f(6) f(6-1) f(6-2) f(5) f(4) 5 3 8

start
definition
Jeff Bivin -- LZHS
6
Write a recursive rule for the sequence
4, 12, 36, 108, 324, . . .
Is it Arithmetic or Geometric?
What is the pattern?
multiply by 3
What is the start?
a1 4
an 3an-1
What is the definition?
7
Write a recursive rule for the sequence
7, 12, 17, 22, 27, . . .
Is it Arithmetic or Geometric?
What is the pattern?
add 5
What is the start?
a1 7
an an-1 5
What is the definition?
8
Write a recursive rule for the sequence
3, 4, 7, 11, 18, 29, 47, . . .
Is it Arithmetic or Geometric?
neither
What is the pattern?
34 7, 4 7 11, 7 11 18
What is the start?
a1 3
a2 4
an an-2 an-1
What is the definition?
9
Find the first three iterates of the function for
the given initial value.
f(x) 5x 3, x0 2
x1 f(x0) f(2) 5(2) 3 13
x2 f(x1) f(13) 5(13) 3 68
x3 f(x2) f(68) 5(68) 3 343
10
Determine the degree of the function
4, 7, 10, 13, 16, 19, 22, 25, 28
3, 3, 3, 3, 3, 3, 3, 3
1st difference
1st Degree
Jeff Bivin -- LZHS
11
Now, write the linear model
1st Degree
f(1)
f(2)
4, 7, 10, 13, 16, 19, 22, 25, 28
(1, 4)
(2, 7)
Jeff Bivin -- LZHS
12
Determine the degreeof the function
-1, 0, 5, 14, 27, 44, 65, 90, 119
1, 5, 9, 13, 17, 21, 25, 29
1st difference
4, 4, 4, 4, 4, 4, 4
2nd difference
2nd Degree
Jeff Bivin -- LZHS
13
Now write the quadratic model
2nd Degree
f(1)
f(2)
f(3)
-1, 0, 5, 14, 27, 44, 65, 90, 119
Solve the system
Jeff Bivin -- LZHS
14
Now write the quadratic model
2nd Degree
f(1)
f(2)
f(3)
-1, 0, 5, 14, 27, 44, 65, 90, 119
a 2
b -5
c 2
Jeff Bivin -- LZHS
15
Determine the degreeof the function
1, 10, 47, 130, 277, 506, 835, 1282, 1865
9, 37, 83, 147, 229, 329, 447, 583
1st difference
28, 46, 64, 82, 100, 118, 136
2nd difference
3rd Degree
18, 18, 18, 18, 18, 18
3rd difference
Jeff Bivin -- LZHS
16
Now write the quadratic model
3rd Degree
f(1)
f(2)
f(3)
f(4)
1, 10, 47, 130, 277, 506, 835, 1282, 1865
Solve the system
Jeff Bivin -- LZHS
17
Now write the quadratic model
3rd Degree
f(1)
f(2)
f(3)
f(4)
1, 10, 47, 130, 277, 506, 835, 1282, 1865
a 3
b -4
c 0
d 2
Jeff Bivin -- LZHS