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 2n-th 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 (compose f g)
    (lambda (x) (f (g x))))

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

(define (repeated op) (compose op op))

((repeated square 2) 5)