What Does This Number Get To?

Algebra Level 3

Where is this number getting to?

  1. Start out with any number 1 < x < 2. 1<x<2.
  2. Set x = 2 − x − 2 . x=2-x^{-2}.
  3. Repeat Step 2 infinitely many times.

Give your answer to 3 decimal places.


The answer is 1.618.

Kim Hyunsoo by Kim Hyunsoo , 17, Philippines

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1 solution

The Value should be the Solution of given equation x 3 + x 2 − 2 = 0 x^{3} +x^{2}-2=0

which has solution as x = 1 x = 1 , 1 2 − ( 5 ) 2 \frac{1}{2}-\frac{\sqrt(5)}{2} and 1 2 + ( 5 ) 2 \frac{1}{2}+\frac{\sqrt(5)}{2}

Only solution lying between the interval ( 1 , 2 ) (1,2) is 1 2 + ( 5 ) 2 \frac{1}{2}+\frac{\sqrt(5)}{2} = 1.618 = 1.618

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