Its minus this time

Algebra Level 4

( 1 − 1 x ) x − 1 = ( 1 − 1 2018 ) 2018 \large \left(1-\frac{1}{x}\right)^{x-1}=\left(1-\frac{1}{2018}\right)^{2018}

Find x x .

Inspiration − - Power of 2000

Bonus: Can you genralize it?


The answer is -2017.

Natalia Dcruz by Natalia Dcruz , 21, India

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1 solution

David Vreken
Feb 17, 2018

Let a = 1 − x a = 1 - x . Then x = 1 − a x = 1 - a and ( 1 − 1 x ) x − 1 = ( 1 − 1 1 − a ) − a = ( − a − a + 1 ) − a = ( a a − 1 ) − a = ( a − 1 a ) a = ( 1 − 1 a ) a (1 - \frac{1}{x})^{x-1} = (1 - \frac{1}{1 - a})^{-a} = (\frac{-a}{-a + 1})^{-a} = (\frac{a}{a - 1})^{-a} = (\frac{a - 1}{a})^a = (1 - \frac{1}{a})^a .

Therefore, in this problem, a = 2018 a = 2018 , and x = 1 − a = 1 − 2018 = − 2017 x = 1 - a = 1 - 2018 = \boxed{-2017} .

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