Beauty of Numbers

Number Theory Level 2

If 2. A B C D E F G H X Y Z ‾ − 1 = 0. A B C D E F G H X Y Z ‾ \overline{2.ABCDEFGHXYZ}^{-1} = \overline{0.ABCDEFGHXYZ} , find A B C D E F G H X Y Z ‾ \overline{ABCDEFGHXYZ} .


The answer is 41421356237.

Arindam Ghosh by Arindam Ghosh , 19, India

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1 solution

Chew-Seong Cheong
Dec 16, 2019

Let x = 0. A B C D E F G H X Y Z ‾ x=\overline{0.ABCDEFGHXYZ} . Then

1 2 + x = x 1 = 2 x + x 2 ⟹ x 2 + 2 x + 1 = 2 ( x + 1 ) 2 = 2 x + 1 = 2 Since x > 0 ⟹ x = 2 − 1 ≈ 0.41421356237 \begin{aligned} \frac 1{2+x} & = x \\ 1 & = 2x + x^2 \\ \implies x^2 + 2x + 1 & = 2 \\ (x+1)^2 & = 2 \\ x+1 & = \sqrt 2 & \small \blue{\text{Since }x>0} \\ \implies x & = \sqrt 2 - 1 \\ & \approx 0.41421356237 \end{aligned}

Therefore, A B C D E F G H X Y Z ‾ = 41421356237 \overline{ABCDEFGHXYZ} = \boxed{41421356237} .

1 Helpful 0 Interesting 0 Brilliant 0 Confused

@Arindam Ghosh , don't put text in LaTex. It is difficult and not a standard in Brilliant.org, not necessary look good too.

Chew-Seong Cheong - 1 year, 5 months ago

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