1. Recursive Functions and Finite Differences
By: Jeffrey Bivin
Lake Zurich High School
Last Updated: November 28, 2005

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
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3. Example 1 start definition f(1) = 5 f(n) = f(n-1) + 10 n = 2
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4. Example 2 start definition f(1) = 3 f(n) = 5•f(n-1) + 2 n = 2
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5. Example 3 start definition 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) = = 2
f(4) = f(4-1) + f(4-2) = f(3) + f(2) = = 3
f(5) = f(5-1) + f(5-2) = f(4) + f(3) = = 5
f(6) = f(6-1) + f(6-2) = f(5) + f(4) = = 8
start
definition
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6. 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
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7. Now, write the linear model
1st Degree
f(1)
f(2)
4, 7, 10, 13, 16, 19, 22, 25, 28
(1, 4)
(2, 7)
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8. Determine the degree of 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
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9. 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
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10. 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

11. Determine the degree of the function
1, 10, 47, 130, 277, 506, 835, 1282, 1865
9, 37, 83, 147, 229, 329, 447,
1st difference
28, 46, 64, 82, 100, 118,
2nd difference
3rd Degree
18, 18, 18, 18, ,
3rd difference
Jeff Bivin -- LZHS

12. 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

13. 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