sicp 1.43

 

Exercise 1.43.  If f is a numerical function and n is a positive integer, then we can form the nth repeated application of f, which is defined to be the function whose value at x is f(f(...(f(x))...)). For example, if f is the function x x + 1, then the nth repeated application of f is the function x x + n. If f is the operation of squaring a number, then the nth repeated application of f is the function that raises its argument to the 2ⁿth power. Write a procedure that takes as inputs a procedure that computes f and a positive integer n and returns the procedure that computes the nth repeated application of f. Your procedure should be able to be used as follows:

 

 

((repeated square 2) 5)
625

 

Hint: You may find it convenient to use compose from exercise 1.42. 

 

(define (repeated f n)
  (if (= n 1)
      f
      (composed f (repeated f (- n 1)))))

(define (composed f g)
  (lambda (x) (f (g x))))

(define (square x)
  (* x x))

((repeated square 2) 5)

 

 

625