Recursive Functions, Iterates, and Finite Differences By: Jeffrey Bivin Lake Zurich High School Last Updated: May 21, 2008

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 a 1 = 5 a n = a n n = 2 a 2 = a (2-1) + 10 a 2 = a a 2 = a 2 = 15 n = 3 a 3 = a (3-1) + 10 a 3 = a a 3 = a 3 = 25 n = 4 a 4 = a (4-1) + 10 a 4 = a a 4 = a 4 = 35 Jeff Bivin -- LZHS

Example 2 f(1) = 3 f(n) = 5f(n-1) + 2 n = 2 f(2) = 5f(2-1) + 2 f(2) = 5f(1) + 2 f(2) = f(2) = 17 n = 3 f(3) = 5f(3-1) + 2 f(3) = 5f(2) + 2 f(3) = f(3) = 87 n = 4 f(4) = 5f(4-1) + 2 f(4) = 5f(3) + 2 f(4) = f(4) = 437 Jeff Bivin -- LZHS

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) = = 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 Jeff Bivin -- LZHS

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? What is the definition? a 1 = 4 a n = 3 ·a n-1

Is it Arithmetic or Geometric? Write a recursive rule for the sequence 7, 12, 17, 22, 27,... What is the pattern? add 5 What is the start? What is the definition? a 1 = 7 a n = a n-1 + 5

Is it Arithmetic or Geometric? Write a recursive rule for the sequence 3, 4, 7, 11, 18, 29, 47,... What is the pattern? 3+4 = 7, = 11, = 18 What is the start? What is the definition? a 1 = 3 a n = a n-2 + a n-1 neither a 2 = 4

Find the first three iterates of the function for the given initial value. f(x) = 5x + 3, x 0 = 2 x 1 = f(x 0 ) = f(2) = 5(2) + 3 = 13 x 2 = f(x 1 ) = f(13) = 5(13) + 3 = 68 x 3 = f(x 2 ) = f(68) = 5(68) + 3 = 343

Determine the degree of the function 4, 7, 10, 13, 16, 19, 22, 25, 28 3, 3, 3, 3, 3, 3, 3, 3 1 st difference Jeff Bivin -- LZHS

Now, write the linear model 4, 7, 10, 13, 16, 19, 22, 25, 28 f(1)f(2) (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 1 st difference 4, 4, 4, 4, 4, 4, 4 2 nd difference Jeff Bivin -- LZHS

Now write the quadratic model -1, 0, 5, 14, 27, 44, 65, 90, 119 f(1)f(2)f(3) Jeff Bivin -- LZHS

Now write the quadratic model -1, 0, 5, 14, 27, 44, 65, 90, 119 f(1)f(2)f(3) Jeff Bivin -- LZHS a = 2 b = -5 c = 2

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

Now write the quadratic model f(1)f(2)f(3) 1, 10, 47, 130, 277, 506, 835, 1282, 1865 f(4) Jeff Bivin -- LZHS

Now write the quadratic model f(1)f(2)f(3) 1, 10, 47, 130, 277, 506, 835, 1282, 1865 f(4) Jeff Bivin -- LZHS a = 3b = -4c = 0d = 2