find-10-1-10-1-10-1-10-1-10-A-5-2-6-B-5-2-6-C-5-2-6-D-none-of-above-please-give-your-answer-and-explain-why-

Question Number 128290 by mr W last updated on 06/Jan/21

f i n d 10 - \frac{1}{10 - \frac{1}{10 - \frac{1}{10 - \frac{1}{10 - \dots \dots}}}} = ?

(A) 5 - 2 \sqrt{6}

(B) 5 + 2 \sqrt{6}

(C) 5 \pm 2 \sqrt{6}

(D) n o n e o f a b o v e

p l e a s e g i v e y o u r a n s w e r a n d e x p l a i n

w h y!

Commented by som(math1967) last updated on 06/Jan/21

I s t h e a n s w a r i s C) 5 \pm 2 \sqrt{6} ?

Commented by som(math1967) last updated on 06/Jan/21

l e t x = 10 - \frac{1}{10 - \frac{1}{10 - \frac{1}{10 - \dots}}}

\Rightarrow x - 10 = - \frac{1}{x}

\Rightarrow x^{2} - 10 x + 1 = 0

x = \frac{10 \pm \sqrt{100 - 4}}{2} = 5 \pm 2 \sqrt{6}

Commented by Dwaipayan Shikari last updated on 06/Jan/21

5 + 2 \sqrt{6} = 10 - (5 - 2 \sqrt{6}) = 10 - \frac{1}{5 + 2 \sqrt{6}}

= 10 - \frac{1}{10 - \frac{1}{5 + 2 \sqrt{6}}} = 10 - \frac{1}{10 - \frac{1}{10 - \frac{1}{5 + 2 \sqrt{6}}}} = 10 - \frac{1}{10 - \frac{1}{10 - \frac{1}{10 - \frac{1}{10 . .}}}}

5 - 2 \sqrt{6} = 10 - (5 + 2 \sqrt{6}) = 10 - \frac{1}{5 - 2 \sqrt{6}} = 10 - \frac{1}{10 - \frac{1}{5 - 2 \sqrt{6}}}

= 10 - \frac{1}{10 - \frac{1}{10 - \frac{1}{10 - \dots}}}

Commented by mr W last updated on 06/Jan/21

5 - 2 \sqrt{6} = \frac{1}{5 + 2 \sqrt{6}}

\Rightarrow \frac{1}{10 - \frac{1}{10 - \frac{1}{10 - \dots}}} = 10 - \frac{1}{10 - \frac{1}{10 - \frac{1}{10 - \dots}}}

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