a_n = a * r_{n − 1}

a_n = r * a_{n − 1}  for every integer n ≥ 1.

#lang racket
;; a - initial value
;; r - common ratio
;; n - number of terms in the sequence


(define py-range                      ; Returns the pythonic `range()`.
  (lambda (start stop (step 1))
    (cond ((>= start stop) '())
          (else (cons (+ (- start 1) step)
                      (py-range (+ start step) stop step))))))

(define geometric-sequence-with-map
  (lambda (a r n)
    (let ((nth-term-geo-seq-closure                 ; explicite formula
             (lambda (k) (* a (expt r (- k 1))))))  ; for kth term
      (map nth-term-geo-seq-closure (py-range 1 (+ n 1))))))

(geometric-sequence-with-map 1/2 1/2 10)
;; '(1/2 1/4 1/8 1/16 1/32 1/64 1/128 1/256 1/512 1/1024)

(define (geometric-sequence a r n)
  (letrec ((k -1)
         (geo-seq (lambda (a r k)
           (cond ((zero? (- n k)) '())
                 (else (cons (* a (expt r (+ k 1)))
                       (geo-seq a r (+ k 1))))))))
    (geo-seq a r k)))

(geometric-sequence 1/2 1/2 10)
;; '(1/2 1/4 1/8 1/16 1/32 1/64 1/128 1/256 1/512 1/1024 1/2048)

(defun py-range (start stop &optional (step 1))
  (loop for x from start below stop by step
        collect x))

(defun geometric-sequence-with-map (a r n)
  (labels ((nth-term-geometric-sequence-closure (k)      ; explicit formula
              (* a (expt r (1- k)))))                    ; for kth term
    (mapcar #'nth-term-geometric-sequence-closure (py-range 1 (1+ n)))))

(geometric-sequence-with-map 1/2 1/2 10)
;; (1/2 1/4 1/8 1/16 1/32 1/64 1/128 1/256 1/512 1/1024)

(defun geometric-sequence (a r n)
  (let ((last-element (* a (expt r (- n 1)))))
    (labels ((geo-seq (a r n &optional (acc-l (list last-element)))
      (cond ((zerop n) acc-l)
            (t (geo-seq a r (- n 1) (cons (* a (expt r (1- n)))
                                          acc-l))))))
       (geo-seq a r n))))

(defun geometric-sequence (a r n)
  (let ((k -1))
    (labels ((geo-seq (a r k &optional acc-l)
                (cond ((zerop (- n k)) acc-l)
                      (t (geo-seq a r (1+ k) (cons (* a (expt r (1+ k)))
                                                   acc-l))))))
      (geo-seq a r k '()))))

(defun geometric-sequence (a r n)
  (let ((k -1))
    (labels ((geo-seq (a r k)
                (cond ((zerop (- n k)) '())
                      (t (cons (* a (expt r (1+ k)))
                               (geo-seq a r (1+ k)))))))
      (geo-seq a r k))))

(geometric-sequence 1/2 1/2 10)
;; (1/2 1/4 1/8 1/16 1/32 1/64 1/128 1/256 1/512 1/1024 1/2048)