My Continued Fraction

Algebra Level 2

x = 2 + 1 2 + 1 2 + 1 2 + … \large x=2+\frac { 1 }{ 2+\frac { 1 }{ 2+\frac { 1 }{2 + \ldots } } }

If x x is the solution to the nested fraction above, and it can be written as a + b a+\sqrt { b }

Express your answer as the sum of a + b a+b .


The answer is 3.

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Isaiah Simeone by isaiah simeone , 23, Australia

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2 solutions

Isaiah Simeone
Sep 25, 2014

x = 2 + 1 2 + 1 2 + 1 . . . . . x=2+\frac { 1 }{ 2+\frac { 1 }{ 2+\frac { 1 }{ ..... } } }

x = 2 + 1 x x=2+\frac { 1 }{ x }

x 2 = 2 x + 1 { x }^{ 2 }=2x+1

x 2 − 2 x − 1 = 0 { x }^{ 2 }-2x-1=0

x = 1 + 2 x=1+\sqrt { 2 }

A n s w e r = 3 Answer = 3

14 Helpful 0 Interesting 0 Brilliant 0 Confused

x 2 − 2 x − 1 = 0 → 1 ± 1 − 1 × − 1 = 1 ± 2 \displaystyle x ^ { 2 } - 2x - 1 = 0 \rightarrow 1 \pm \sqrt { 1 - 1 \times -1 } = 1 \pm \sqrt { 2 } .

Not x = 1 + 2 x = 1 + \sqrt { 2 } .

. . - 2 months, 2 weeks ago

I substituted it with the even quadratic formula.

. . - 2 months, 2 weeks ago
Curtis Clement
Feb 20, 2015

1 x − 2 = x ⇒ x 2 − 2 x − 1 = 0 \frac{1}{x-2} = x \Rightarrow\ x^2 -2x-1 =0 ( x − 1 ) 2 = 2 (x-1)^2 =2 N o w x > 0 s o \large Now \ x > 0 \ so x − 1 = + 2 x-1 = +\sqrt{2} ∴ x = 1 + 2 \therefore\ x = 1 + \sqrt{2} a n s w e r = 3 \large answer =3

1 Helpful 0 Interesting 0 Brilliant 0 Confused

There is no proof that x x must be positive.

. . - 2 months, 2 weeks ago

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