Exercise 1.43

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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.