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!


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

$f i n d 10 - \frac{1}{10 - \frac{1}{10 - \frac{1}{10 - \frac{1}{10 - . . . . . .}}}} = ?$ $(A) 5 - 2 \sqrt{6}$ $(B) 5 + 2 \sqrt{6}$ $(C) 5 \pm 2 \sqrt{6}$ $(D) n o n e o f a b o v e$ $p l e a s e g i v e y o u r a n s w e r a n d e x p l a i n$ $w h y!$
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

$l e t x = 10 - \frac{1}{10 - \frac{1}{10 - \frac{1}{10 - . . .}}}$ $\Rightarrow x - 10 = - \frac{1}{x}$ $\Rightarrow x^{2} - 10 x + 1 = 0$ $x = \frac{10 \pm \sqrt{100 - 4}}{2} = 5 \pm 2 \sqrt{6}$
Commented by Dwaipayan Shikari last updated on 06/Jan/21

$5 + 2 \sqrt{6} = 10 - (5 - 2 \sqrt{6}) = 10 - \frac{1}{5 + 2 \sqrt{6}}$ $= 10 - \frac{1}{10 - \frac{1}{5 + 2 \sqrt{6}}} = 10 - \frac{1}{10 - \frac{1}{10 - \frac{1}{5 + 2 \sqrt{6}}}} = 10 - \frac{1}{10 - \frac{1}{10 - \frac{1}{10 - \frac{1}{10 . .}}}}$ $5 - 2 \sqrt{6} = 10 - (5 + 2 \sqrt{6}) = 10 - \frac{1}{5 - 2 \sqrt{6}} = 10 - \frac{1}{10 - \frac{1}{5 - 2 \sqrt{6}}}$ $= 10 - \frac{1}{10 - \frac{1}{10 - \frac{1}{10 - . . .}}}$
Commented by mr W last updated on 06/Jan/21

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