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a4.scm
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a4.scm
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"problem 1. (a)"
(define (lucas n) (cond ((= n 0) 2)
((= n 1) 1)
((> n 1) (+ (lucas (- n 1))
(lucas (- n 2))))))
;(lucas 30)
;(lucas 35)
;(lucas 40)
"problem 1. (b)"
(define (fast-Lucas-help n k lucas-a lucas-b)
(if (= n k)
lucas-a
(fast-Lucas-help n (+ k 1) (+ lucas-a lucas-b) lucas-a )))
(define (fast-Lucas n) (fast-Lucas-help n 1 1 2))
" Recursive calls Recursive calls made by"
" made by (Lucas k) (fast-Lucas-help k 1 1 2)"
"k=1 0 0"
"k=2 2 1"
"k=3 4 2"
"k=4 8 3"
"k=5 16 4"
"k=6 24 5"
"problem 2. (a)"
(define (sum term a next b)
(if (> a b)
0
(+ (term a)
(sum term (next a) next b))))
(define (harmonic n) (sum (lambda (x) (/ 1 x)) 1 (lambda (x) (+ x 1)) n))
"problem 2. (b)"
(define (sum-i term a next b accumulator)
(if (> a b)
accumulator
(sum-i term (next a) next b (+ accumulator (term a)))))
(define (harmonic-i n) (sum-i (lambda (x) (/ 1 x)) 1 (lambda (x) (+ x 1)) n 0))
"problem 2. (c)"
(harmonic 1)
(harmonic-i 1)
(harmonic 50)
(harmonic-i 50)
(harmonic 100)
(harmonic-i 100)
"problem 3."
(define (compose f g) (lambda (x) (f (g x))))
((compose (lambda (x) (* x x)) (lambda (x) (+ x 1))) 6)
"problem 4."
(define (repeated f n)
(if (= n 0)
(lambda (x) x)
(compose (repeated f (- n 1)) f)))
"problem 5."
(define (m91 x)
(if (> x 100)
(- x 10)
((repeated m91 91) (+ x 901))))