perplexus.info :: Sequences : Initial Term Identification

Initial Term Identification (Posted on 2015-07-21)

A sequence {X₀, X₁, X₂, ...., X_n} is given by:
X_n(1 - X_n-1) = X_n-1

Find X₀, given that: X₂₀₁₅ = 2015

No Solution Yet Submitted by K Sengupta

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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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