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Question Number 79377 by mr W last updated on 24/Jan/20

Commented by mr W last updated on 24/Jan/20

$t h e s i d e l e n g t h o f t h e s q u a r e i s a .$ $f i n d t h e r a d i u s o f t h e n - t h s m a l l$ $c i r c l e r_{n} .$
Commented by key of knowledge last updated on 24/Jan/20

$mr w you like this problem!$
Commented by mr W last updated on 24/Jan/20

$y e s s i r . i t r y t o l e a r n s o m e n e w t h i n g s$ $t h r o u g h t h i s p r o b l e m .$
	Answered by mr W last updated on 25/Jan/20

	$w e h a v e g o t r_{0} = \frac{a}{3} (v e r t i c a l s e m i c i r c l e)$ $i f w e k n o w r_{n}, h o w c a n w e g e t r_{n + 1} ?$ $w e c a n a p p l y t h e {D e s c a r t e s}^{'} t h e o r e m$ $a s f o l l o w i n g :$ ${(- \frac{1}{a} + \frac{2}{a} + \frac{1}{r_{n}} + \frac{1}{r_{n + 1}})}^{2} = 2 (\frac{1}{a^{2}} + \frac{4}{a^{2}} + \frac{1}{r_{n}^{2}} + \frac{1}{r_{n + 1}^{2}})$ $\frac{2}{{a r}_{n}} + \frac{2}{{a r}_{n + 1}} + \frac{2}{r_{n} r_{n + 1}} = \frac{9}{a^{2}} + \frac{1}{r_{n}^{2}} + \frac{1}{r_{n + 1}^{2}}$ ${(\frac{a}{r_{n + 1}} - \frac{a}{r_{n}})}^{2} - 2 (\frac{a}{r_{n + 1}} - \frac{a}{r_{n}}) + 9 - 4 \frac{a}{r_{n}} = 0$ $\Rightarrow \frac{a}{r_{n + 1}} - \frac{a}{r_{n}} = 1 + 2 \sqrt{\frac{a}{r_{n}} - 2}$ $\Rightarrow \frac{a}{r_{n + 1}} = \frac{a}{r_{n}} + 2 \sqrt{\frac{a}{r_{n}} - 2} + 1$ $\frac{a}{r_{0}} = 3$ $\frac{a}{r_{n + 1}} - 2 = \frac{a}{r_{n}} - 2 + 2 \sqrt{\frac{a}{r_{n}} - 2} + 1$ $\frac{a}{r_{n + 1}} - 2 = {(\sqrt{\frac{a}{r_{n}} - 2} + 1)}^{2}$ $l e t b_{n} = \sqrt{\frac{a}{r_{n}} - 2}$ $\Rightarrow b_{n + 1}^{2} = {(b_{n} + 1)}^{2}$ $\Rightarrow b_{n + 1} = b_{n} + 1 \Rightarrow {i t}^{'} s a A . P .!$ $\Rightarrow b_{n} = b_{0} + n$ $b_{0} = \sqrt{\frac{a}{r_{0}} - 2} = \sqrt{3 - 2} = 1$ $\Rightarrow b_{n} = n + 1$ $\Rightarrow \sqrt{\frac{a}{r_{n}} - 2} = n + 1$ $\Rightarrow \frac{a}{r_{n}} = {(n + 1)}^{2} + 2$ $\Rightarrow r_{n} = \frac{a}{{(n + 1)}^{2} + 2}$
	Commented by behi83417@gmail.com last updated on 25/Jan/20

	$great! dear master .$ $You can't use 'macro parameter character #' in math mode$ $(and also meet a nostalgia!)$ $[nostos : return to home$ $algia : be in suffer]$
	Commented by mr W last updated on 25/Jan/20

	$h a h a, y o u s t i l l r e m e m b e r t h a t o l d i e!$ $l e t m e t r y i f i t c o u l d b e s o l v e d i n$ $s i m i l a r y w a y .$
	Commented by behi83417@gmail.com last updated on 25/Jan/20

	$thanks in advance dear master .$
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