This problem defines the Calkin-Wilf-Newman sequence of positive rational numbers. The

Chapter 9, Problem 60

(choose chapter or problem)

This problem defines the Calkin-Wilf-Newman sequence of positive rational numbers. The sequence is remarkable because every positive rational number appears as one of its terms and none appears more than once. Every real number x can be written as an integer A plus a number B where 0 B < 1. For example, for x = 12/5 = 2+2/5 we have A = 2 and B = 2/5. For x = 3= 3+0 we have A = 3 and B = 0. Define the function f(x) by f(x) = A + (1 B). For example, f(12/5) = 2 + (1 2/5) = 13/5 and f(3) = 3 + (1 0) = 4. (a) Evaluate f(x) for x = 25/8, 13/9, and . (b) Find the first six terms of the recursively defined Calkin-Wilf-Newman sequence: sn = 1/f(sn1) for n > 1 and s1 = 1.