Continued fractions or continued fun?

Algebra Level 4

1 3 + 1 2 + 1 1 + 1 3 + 1 2 + 1 1 + 1 ⋱ \large \cfrac{1}{3 + \cfrac{1}{2 + \cfrac{1}{1 + \cfrac{1}{3 + \cfrac{1}{2 + \cfrac{1}{1 + \cfrac{1}{\ddots }}}}}}}

Evaluate the infinitely nested function above. Give your answer to 3 decimal places.

Clarification : The 3 , 2 , 1 , 3 , 2 , 1 , 3 , 2 , 1 , … 3,2,1,3,2,1,3,2,1,\ldots pattern in repeating.


The answer is 0.29753.

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Rohit Udaiwal by Rohit Udaiwal , 20, India

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1 solution

Akhil Bansal
Dec 10, 2015

Let x \color{#3D99F6}{x} be equal to the given expression.

x = 1 3 + 1 2 + 1 1 + x \large \color{#3D99F6}{x}= \cfrac{1}{3 + \cfrac{1}{2 + \cfrac{1}{1 + \color{#3D99F6}{x}}}} x = 1 3 + 1 + x 3 + 2 x \large \color{#3D99F6}{x}= \cfrac{1}{3 + \cfrac{1+\color{#3D99F6}{x}}{3 + 2\color{#3D99F6}{x}}} x = 3 + 2 x 10 + 7 x \large \color{#3D99F6}{x} = \dfrac{3 + 2\color{#3D99F6}{x}}{10 + 7\color{#3D99F6}{x}} 7 x 2 + 8 x − 3 = 0 \large 7\color{#3D99F6}{x}^2 + 8\color{#3D99F6}{x} - 3 = 0

By the Quadratic Formula , x = 0.298 , − 1.440 \large \color{#3D99F6}{x} = 0.298,-1.440 Because given expression constains only positive terms.Hence,neglecting − 1.440 -1.440

24 Helpful 0 Interesting 0 Brilliant 0 Confused

I divided by a instead of 2a.. sad

Nanda Rahsyad - 5 years, 6 months ago

I didnt get how you got 3+2x in the third line

Mr Yovan - 5 years, 6 months ago

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Just take the L.C.M
2 + 1 1 + x = 2 ( 1 + x ) + 1 1 + x = 3 + 2 x 1 + x 2 + \dfrac{1}{1+x} = \dfrac{2(1+x) + 1}{1+x} = \dfrac{3+2x}{1+x}

Akhil Bansal - 5 years, 6 months ago

Also can 1/(1/(1/x-3)-2)-1=x so 1/(x/(1-3x)-2)-1=x (1-3x)/(7x-2)-1=x 3-10x=7x²-2x So 7x²+8x-3=0 Positive solution is x=(-4+√37)/7~0.298

Nikola Djuric - 5 years, 6 months ago

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