1.43.scm

;;;; 1.43
;; If $f$ is a numerical function and $n$ is a positive integer, then we can form the $n$th 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 $x \rightarrow x + 1$, then the $n$th repeated application of $f$ is the function $x \rightarrow x + n$. If $f$ is the operation of squaring a number, then the $n$th repeated application of $f$ is the function that raises its argument to the $2^n$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 $n$th 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.

;;; Answer
(load "1.42.scm")
(define (repeated f n)
  (define (iter i p)
    (if (= i n)
        p
        (iter (+ i 1) (compose f p))))
  (iter 1 f))

;; ;; Test
;; (load "0.util.scm")
;; (define (square x) (expt x 2))
;; (test ((repeated square 2) 5) (expt 5 4))