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A sequence \((a_n)\) is defined as follows: \(a_1 = 1,a_2 = \frac{1}{2},\)  and \(a_n = \frac{1 - a_{n - 1}}{2a_{n - 2}} \)
for all \(n \ge 2\) Find \(a_{120}\)

 Jun 14, 2019
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I had the computer list them for you:

 

(1, 1/2, 1/4, 3/4, 1/2, 1/3, 2/3, 1/2, 3/8, 5/8, 1/2, 2/5, 3/5, 1/2, 5/12, 7/12, 1/2, 3/7, 4/7, 1/2, 7/16, 9/16, 1/2, 4/9, 5/9, 1/2, 9/20, 11/20, 1/2, 5/11, 6/11, 1/2, 11/24, 13/24, 1/2, 6/13, 7/13, 1/2, 13/28, 15/28, 1/2, 7/15, 8/15, 1/2, 15/32, 17/32, 1/2, 8/17, 9/17, 1/2, 17/36, 19/36, 1/2, 9/19, 10/19, 1/2, 19/40, 21/40, 1/2, 10/21, 11/21, 1/2, 21/44, 23/44, 1/2, 11/23, 12/23, 1/2, 23/48, 25/48, 1/2, 12/25, 13/25, 1/2, 25/52, 27/52, 1/2, 13/27, 14/27, 1/2, 27/56, 29/56, 1/2, 14/29, 15/29, 1/2, 29/60, 31/60, 1/2, 15/31, 16/31, 1/2, 31/64, 33/64, 1/2, 16/33, 17/33, 1/2, 33/68, 35/68, 1/2, 17/35, 18/35, 1/2, 35/72, 37/72, 1/2, 18/37, 19/37, 1/2, 37/76, 39/76, 1/2, 19/39, 20/39, 1/2, 39/80, 41/80, 1/2, 20/41)

P.S. I may have misread your question, so be careful with this result. 

 Jun 14, 2019

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