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)
  (lambda (x)
    (define (iter result i)
      (if (= n i)
      (f result)
      (iter (f result) (inc i))))
    (iter x 1)))
 
((repeated square 2) 5)

(define (repeated f n)
  (if (= n 0)
      (lambda (x) x)
      (lambda (x) ((compose f (repeated f (- n 1))) x))))

And one more, not using compose:

(define (repeated f n)
  (if (= n 0)
      (lambda (x) x)
      (lambda (x) (f ((repeated f (- n 1)) x)))))