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Initial Term Identification (Posted on 2015-07-21) Difficulty: 3 of 5
A sequence {X0, X1, X2, ...., Xn} is given by:
Xn(1 - Xn-1) = Xn-1

Find X0, given that: X2015 = 2015

No Solution Yet Submitted by K Sengupta    
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Solution Solution (spoiler alert) Comment 3 of 3 |
Rearranging the given equation: x_(n-1) = 1 / (1 + 1/x_n )
So, x_(n-2) = 1 / (1 + 1/x_(n-1) ) 
                 = 1 / (1 + 1 + 1/x_n) 
                = 1 / (2+1/x_n)
Similarly, x_(n-3) = 1 / (3 + 1/x_n)
and the pattern becomes clear: x_(n-j) = 1 / (j + 1/x_n)

Putting j=2015: x_0 = 1 / (2015 + 1/ x_2015)
                             = 1/ (2015 + 1/2015)
                             = 2015 / (2015^2 + 1) 

Nice prob, KS!

  Posted by JayDeeKay on 2015-07-21 20:32:46
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