Recursive Functions and Finite Differences By: Jeffrey Bivin Lake Zurich High School jeff.bivin@lz95.org Last Updated: November 28, 2005

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

Example 1 start definition f(1) = 5 f(n) = f(n-1) + 10 n = 2 Jeff Bivin -- LZHS

Example 2 start definition f(1) = 3 f(n) = 5•f(n-1) + 2 n = 2 Jeff Bivin -- LZHS

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) = 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

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

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

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 Jeff Bivin -- LZHS

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

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

Determine the degree of 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

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

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