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Complex Ratio and Number Resolution (Posted on 2016-04-13) Difficulty: 3 of 5
Consider the function:
G(Z) = (Z+i)/(Z-i) for all complex numbers Z ≠ i, and:
The sequence {Zn} is defined as:
G(Zn-1) = Zn, whenever n is a positive integer, and:
Z0 = 1/137 + i

Find each of A, B, C and D, given that:
Z2014 = A+B*i, and:
Z2015 = C+D*i

No Solution Yet Submitted by K Sengupta    
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Solution computer assisted solution | Comment 1 of 2
   10   Real=1//137:Imag=1
   20   for I=1 to 2015
   30     a=real:c=real
   40     b=imag+1:d=imag-1
   50     real=(a*c+b*d)//(c*c+d*d)
   60     imag=(b*c-a*d)//(c*c+d*d)
   70     if i=2014 or i=2015 then
   80       :print i,real;imag
   90    next   
   
Uses symbols a,b,c and d as in Wolfram Mathworld's article on complex division.


It finds

2014:  1 + 274*i
2015:  37538/37265 + 1/37265 * i

In fact the 2014th value is preceded by 

2013:  1/137 + i

so actually it has been a 3-cycle since:

0:   1/137 + i
1:   1 + 274*i
2:   37538/37265 + 1/37265 * i
3:   1/137 + i

  Posted by Charlie on 2016-04-13 14:37:15
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