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

((repeated square 2) 5) ;Value: 625