Find the exact value

Algebra Level 3

$\huge{2+\frac{1}{2+\frac{1}{2+...}}= \ ?}$

None of answers shown is correct

-0.41421356237

1+\sqrt{2}

1-\sqrt{2}

There is no fix value

2.41421356237

1 solution

Vu Hoang
Dec 19, 2015

Set $2+\frac{1}{2+\frac{1}{2+...}}=x$ , so $x>=2$ .Then $\frac{1}{2+\frac{1}{2+...}}=x-2$ .Flip 2 sides over we got $2+\frac{1}{2+...}=\frac{1}{x-2}$ .Realize the left side of equation is definitely x .Hence, $x=\frac{1}{x-2}$ .We got quadratic equation $x^2-2x-1=0$ .The solution will be $1+\sqrt(2)$

Find the exact value

1 solution

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