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Question Number 162530 by mr W last updated on 30/Dec/21

Commented by mr W last updated on 30/Dec/21

$f i n d t h e s u m o f a r e a s o f a l l (i n f i n i t e)$ $r e d c i r c l e s$
Commented by aleks041103 last updated on 30/Dec/21

$T h i s i s a v e r y d i f f i c u l t q u e s t i o n .$ $U p u n t i l l n o w I c o u l d o n l y p r o v e t h a t$ $t h e c e n t e r s l i e o n a p a r a b o l a .$
Commented by mr W last updated on 30/Dec/21

$i n d e e d {i t}^{'} s v e r y h a r d, p e r h a p s n o t$ $s o l v a b l e i n a n e x a c t w a y .$
Commented by aleks041103 last updated on 30/Dec/21

$I f o u n d t h i s r e c u r s i v e r e l a t i o n :$ $r_{i + 1} = \frac{r_{i}}{{(2 r_{i} + 1)}^{2}} (3 - 2 r_{i} - 2 \sqrt{2 - 4 r_{i}})$
Commented by mr W last updated on 30/Dec/21

$i g o t t h i s t o o .$ $\frac{λ_{n + 1}}{λ_{n}} = \frac{3 - 2 λ_{n} - 2 \sqrt{2 (1 - 2 λ_{n})}}{{(2 λ_{n} + 1)}^{2}}$ $w i t h λ_{n} = \frac{r_{n}}{R}, R = r a d i u s o f s e m i c i r c l e = 1$ $r_{0} = \frac{1}{2}$ $r_{1} = \frac{1}{4}$ $r_{2} = \frac{1}{18}$ $r_{3} = \frac{1}{100}$ $r_{4} = \frac{1}{578}$ $r_{5} = \frac{1}{3 364}$ $r_{6} = \frac{1}{19 602}$ $r_{7} = \frac{1}{114 244}$ $r_{8} = \frac{1}{665 858}$ $r_{9} = \frac{1}{3 880 900}$ $r_{10} = \frac{1}{22 619 538}$ $r_{11} = \frac{1}{131 836 324}$ $r_{12} = \frac{1}{768 398 402}$ $r_{13} = \frac{1}{4 478 554 084}$ $r_{14} = \frac{1}{26 102 926 098}$ $r_{15} = \frac{1}{152 139 002 500}$ $r_{16} = \frac{1}{886 731 088 898}$ $. . .$
Commented by Tawa11 last updated on 30/Dec/21

$Great sir . Please show us the prove sir .$
Commented by mr W last updated on 30/Dec/21

$i t s e e m s t h a t \frac{1}{r_{n}} i s a l w a y s a n i n t e g e r .$
Commented by aleks041103 last updated on 30/Dec/21

$I t r i e d u p t o n = 100, a n d i n d e e d \frac{1}{r_{n}} r e m a i n s$ $a n i n t e g e r .$
Commented by peter frank last updated on 31/Dec/21

$great mind think a like$
Commented by mr W last updated on 01/Jan/22

$a s w e c a n s e e l a t e r,$ $\frac{1}{r_{n}} = \underset{k = 0}{\sum^{n}} 2^{k} C_{2 k}^{2 n} + 1$ $w h i c h i s i n d e e d a l w a y s i n t e g e r .$
	Answered by mr W last updated on 30/Dec/21

	Commented by mr W last updated on 02/Jan/22

	$s a y t h e r a d i u s o f t h e s e m i c i r c l e i s R .$ $s a y t h e b i g g e s t i n s c r i b e d c i r c l e i n t h e$ $m i d d l e o f s e m i c i r c l e h a s t h e r a d u s r_{0}$ $w h i c h i s \frac{R}{2} .$ $s a y t h e c e n t e r o f n^{t h} i n s c r i b e d c i r c l e$ $i s A a n d i t s r a d i u s i s r_{n} .$ $s a y t h e c e n t e r o f {(n + 1)}^{t h} i n s c r i b e d$ $c i r c l e i s B a n d i t s r a d i u s i s r_{n + 1} .$ $O A = R - r_{n}$ $O B = R - r_{n + 1}$ $O C = \sqrt{{(R - r_{n})}^{2} - r_{n}^{2}} = \sqrt{R (R - 2 r_{n})}$ $O D = \sqrt{{(R - r_{n + 1})}^{2} - r_{n + 1}^{2}} = \sqrt{R (R - 2 r_{n + 1})}$ $A B = r_{n} + r_{n + 1}$ ${(r_{n} + r_{n + 1})}^{2} - {(r_{n} - r_{n + 1})}^{2} = {(\sqrt{R (R - 2 r_{n})} + \sqrt{R (R - 2 r_{n + 1})})}^{2}$ $2 r_{n} r_{n + 1} = R^{2} - R (r_{n} + r_{n + 1}) + R \sqrt{(R - 2 r_{n}) (R - 2 r_{n + 1})}$ $l e t λ_{n} = \frac{r_{n}}{R}$ $2 λ_{n} λ_{n + 1} + λ_{n} + λ_{n + 1} - 1 = \sqrt{(1 - 2 λ_{n}) (1 - 2 λ_{n + 1})}$ $\begin{matrix} 4 λ_{n}^{2} λ_{n + 1}^{2} + {(λ_{n} + λ_{n + 1})}^{2} + 4 λ_{n} λ_{n + 1} (λ_{n} + λ_{n + 1} - 2) = 0 \end{matrix} . . . (I)$ ${(2 λ_{n} + 1)}^{2} {(\frac{λ_{n + 1}}{λ_{n}})}^{2} - 2 (3 - 2 λ_{n}) (\frac{λ_{n + 1}}{λ_{n}}) + 1 = 0$ $\frac{λ_{n + 1}}{λ_{n}} = \frac{3 - 2 λ_{n} - \sqrt{{(3 - 2 λ_{n})}^{2} - {(2 λ_{n} + 1)}^{2}}}{{(2 λ_{n} + 1)}^{2}}$ $\frac{λ_{n + 1}}{λ_{n}} = \frac{3 - 2 λ_{n} - 2 \sqrt{2 (1 - 2 λ_{n})}}{{(2 λ_{n} + 1)}^{2}}$ $\frac{λ_{n + 1}}{λ_{n}} = \frac{2 + 1 - 2 λ_{n} - 2 \sqrt{2 (1 - 2 λ_{n})}}{{(2 - (1 - 2 λ_{n}))}^{2}}$ $\frac{λ_{n + 1}}{λ_{n}} = \frac{{(\sqrt{2} - \sqrt{1 - 2 λ_{n}})}^{2}}{{((\sqrt{2}) - {(\sqrt{1 - 2 λ_{n}})}^{2})}^{2}}$ $\frac{λ_{n + 1}}{λ_{n}} = \frac{1}{{(\sqrt{2} + \sqrt{1 - 2 λ_{n}})}^{2}}$ $\frac{1}{λ_{n + 1}} = \frac{{(\sqrt{2} + \sqrt{1 - 2 λ_{n}})}^{2}}{λ_{n}} = \frac{3 - 2 λ_{n} + 2 \sqrt{2 (1 - 2 λ_{n})}}{λ_{n}}$ $\Rightarrow \frac{1}{λ_{n + 1}} = \frac{3 + 2 \sqrt{2 (1 - 2 λ_{n})}}{λ_{n}} - 2 . . . (i)$ $w e c a n t r a n s f o r m e q n . (I) a l s o i n t o$ ${(2 λ_{n + 1} + 1)}^{2} {(\frac{λ_{n}}{λ_{n + 1}})}^{2} - 2 (3 - 2 λ_{n + 1}) (\frac{λ_{n}}{λ_{n + 1}}) + 1 = 0$ $\frac{λ_{n}}{λ_{n + 1}} = \frac{3 - 2 λ_{n + 1} + \sqrt{{(3 - 2 λ_{n + 1})}^{2} - {(2 λ_{n + 1} + 1)}^{2}}}{{(2 λ_{n + 1} + 1)}^{2}}$ $s i m i l a r l y t o a b o v e,$ $\frac{λ_{n}}{λ_{n + 1}} = \frac{1}{{(\sqrt{2} - \sqrt{1 - 2 λ_{n + 1}})}^{2}}$ $\frac{1}{λ_{n}} = \frac{{(\sqrt{2} - \sqrt{1 - 2 λ_{n + 1}})}^{2}}{λ_{n + 1}} = \frac{3 - 2 λ_{n + 1} - 2 \sqrt{2 (1 - 2 λ_{n + 1})}}{λ_{n + 1}}$ $\frac{1}{λ_{n}} = \frac{3 - 2 \sqrt{2 (1 - 2 λ_{n + 1})}}{λ_{n + 1}} - 2$ $o r$ $\Rightarrow \frac{1}{λ_{n - 1}} = \frac{3 - 2 \sqrt{2 (1 - 2 λ_{n})}}{λ_{n}} - 2 . . . (i i)$ $(i) + (i i) :$ $\frac{1}{λ_{n + 1}} + \frac{1}{λ_{n - 1}} = \frac{6}{λ_{n}} - 4$ $\begin{matrix} \frac{1}{λ_{n + 1}} - \frac{6}{λ_{n}} + \frac{1}{λ_{n - 1}} + 4 = 0 \end{matrix}$ $l e t \frac{1}{λ_{n}} = ξ_{n} + c$ $(ξ_{n + 1} + c) - 6 (ξ_{n} + c) + (ξ_{n - 1} + c) + 4 = 0$ $ξ_{n + 1} - 6 ξ_{n} + ξ_{n - 1} + 4 - 4 c = 0$ $s e t 4 - 4 c = 0, i . e . c = 1$ $\Rightarrow ξ_{n + 1} - 6 ξ_{n} + ξ_{n - 1} = 0$ $x^{2} - 6 x + 1 = 0 (c h a r a c t e r i s t i c e q n .)$ $\Rightarrow x = 3 \pm 2 \sqrt{2} = {(\sqrt{2} \pm 1)}^{2}$ $\Rightarrow ξ_{n} = A {(\sqrt{2} + 1)}^{2 n} + B {(\sqrt{2} - 1)}^{2 n}$ $\frac{1}{λ_{n}} = A {(\sqrt{2} + 1)}^{2 n} + B {(\sqrt{2} - 1)}^{2 n} + 1$ $λ_{0} = \frac{1}{2}$ $2 = A + B + 1 \Rightarrow A + B = 1$ $λ_{1} = λ_{- 1} d u e t o s y m m e t r y$ $A {(\sqrt{2} + 1)}^{2} + B {(\sqrt{2} - 1)}^{2} + 1 = A {(\sqrt{2} + 1)}^{- 2} + B {(\sqrt{2} - 1)}^{- 2} + 1$ $A {(\sqrt{2} + 1)}^{2} + B {(\sqrt{2} - 1)}^{2} = A {(\sqrt{2} - 1)}^{2} + B {(\sqrt{2} + 1)}^{2}$ $(A - B) {(\sqrt{2} + 1)}^{2} = (A - B) {(\sqrt{2} - 1)}^{2}$ $\Rightarrow A = B \Rightarrow A = B = \frac{1}{2}$ $\begin{matrix} \frac{1}{λ_{n}} = \frac{{(\sqrt{2} + 1)}^{2 n} + {(\sqrt{2} - 1)}^{2 n}}{2} + 1 \end{matrix}$ $\Rightarrow \frac{1}{λ_{n}} = \frac{{(\sqrt{2} + 1)}^{2 n} + {(\sqrt{2} + 1)}^{- 2 n} + 2}{2}$ $\Rightarrow \frac{1}{λ_{n}} = 2 \times {[\frac{{(\sqrt{2} + 1)}^{n} + {(\sqrt{2} + 1)}^{- n}}{2}]}^{2}$ $\Rightarrow \frac{1}{λ_{n}} = 2 \times {[\frac{e^{n \ln (\sqrt{2} + 1)} + e^{- n \ln (\sqrt{2} + 1)}}{2}]}^{2}$ $\Rightarrow \frac{1}{λ_{n}} = 2 \cosh^{2} [n \ln (\sqrt{2} + 1)]$ $\begin{matrix} r_{n} = \frac{1}{2 \cosh^{2} [n \ln (\sqrt{2} + 1)]} \end{matrix}$ $t o t a l a r e a o f a l l r e d c i r c l e s :$ $A = π r_{0}^{2} + 2 \underset{n = 1}{\sum^{\infty}} π r_{n}^{2}$ $A = \frac{π}{4} + \frac{π}{2} \underset{n = 1}{\sum^{\infty}} \frac{1}{\cosh^{4} [n \ln (\sqrt{2} + 1)]}$ $\approx 1.198 137 490 (76 % o f s e m i c i r c l e)$ $\underset{―}{a d d i t i o n a l n o t e :}$ $w e c a n a l s o p r o c e e d a s f o l l o w i n g :$ $\frac{1}{r_{n}} = \frac{{(\sqrt{2} + 1)}^{2 n} + {(\sqrt{2} - 1)}^{2 n}}{2} + 1$ $\frac{1}{r_{n}} = \frac{\underset{k = 0}{\sum^{2 n}} C_{k}^{2 n} {(\sqrt{2})}^{k} + \underset{k = 0}{\sum^{2 n}} C_{k}^{2 n} {(\sqrt{2})}^{k} {(- 1)}^{2 n - k}}{2} + 1$ $\frac{1}{r_{n}} = \underset{k = 0}{\sum^{n}} 2^{k} C_{2 k}^{2 n} + 1$ $t h i s i s c l e a r l y a l w a y s a n e v e n i n t e g e r .$ $[c o m p l e t e d o n 01 / 01 / 2022, 2 : 00 h]$
	Commented by Tawa11 last updated on 30/Dec/21

	$Wow, thanks sir . God bless you more .$
	Commented by Tawa11 last updated on 30/Dec/21

	$Thanks for your time sir$
	Commented by ajfour last updated on 22/Jan/22

	$r_{n + 1}^{2} + x_{n + 1}^{2} = {(R - r_{n + 1})}^{2}$ ${(x_{n + 1} - x_{n})}^{2} = 4 r_{n} r_{n + 1}$ $\Rightarrow {(\sqrt{R^{2} - 2 {R r}_{n + 1}} - \sqrt{R^{2} - 2 {R r}_{n}})}^{2}$ $= 4 r_{n} r_{n + 1}$ $\Rightarrow 2 R^{2} - 2 R (r_{n} + r_{n + 1}) - 4 r_{n} r_{n + 1}$ $= 2 \sqrt{(R^{2} - 2 {R r}_{n}) (R^{2} - 2 {R r}_{n + 1})}$ $s a y R = 1 \Rightarrow$ ${(1 - r_{n} - r_{n + 1} - 2 r_{n} r_{n + 1})}^{2}$ $= 1 + 4 r_{n} r_{n + 1} - 2 (r_{n} + r_{n + 1})$ $\Rightarrow {(r_{n} + r_{n + 1})}^{2} + 4 r_{n}^{2} r_{n + 1}^{2}$ $+ 4 r_{n} r_{n + 1} (r_{n} + r_{n + 1}) = 4 r_{n} r_{n + 1}$ $\Rightarrow$ $(1 + 4 r_{n} + 4 r_{n}^{2}) r_{n + 1}^{2}$ $+ (2 r_{n} + 4 r_{n}^{2} - 4 r_{n}) r_{n + 1} + r_{n}^{2} = 0$ $. . .$
	Answered by aleks041103 last updated on 30/Dec/21

	Commented by aleks041103 last updated on 30/Dec/21

	$A s s e e n i n t h e i m a g e P H = P N = r$ $\Rightarrow O P + P H = O P + P N = O N = R = c o n s t .$ $\Rightarrow O P + P H = c o n s t .$ $\Rightarrow P l i e s o n a p a r a b o l a w i t h f o c u s a t O$
	Commented by aleks041103 last updated on 30/Dec/21

	${t h a t}^{'} s j u s t i n t e r e s t i n g t o n o t e .$
	Commented by aleks041103 last updated on 31/Dec/21

	$S i n c e i t i s a p a r a b o l a w i t h f o c u s a t O$ $f (x) = {a x}^{2} + b$ $f (\pm 1) = a + b = 0$ $f (0) = r_{0} = \frac{1}{2} = b$ $\Rightarrow f (x) = \frac{1 - x^{2}}{2}$
	Commented by mr W last updated on 31/Dec/21

	$a l e x s i r :$ $y e s, t h e l o c u s o f t h e c e n t e r o f a c i r c l e$ $i n s c r i b e d i n a c i r c u l a r s e g m e n t i s$ $a p a r a b o l a .$
	Commented by mr W last updated on 31/Dec/21

	Commented by aleks041103 last updated on 31/Dec/21

	$m r W, h o w a r e y o u m a k i n g t h o s e d r a w i n g s ?$
	Commented by mr W last updated on 22/Jan/22

	$t h e d i a g r a m i s m a d e w i t h$ $t h e A p p L E H K D I A G R A M$ $(r e c o m m e n d e d b y A j f o u r s i r)$
	Answered by aleks041103 last updated on 31/Dec/21

	$I f o u n d a r e c u r s i v e f o r m u l a f o r \frac{1}{r_{i}}$ $\frac{1}{r_{i + 2}} = \frac{6}{r_{i + 1}} - \frac{1}{r_{i}} - 4$ $t h e p r o o f i s w a a y t o l o n g, s o I^{'} m g o i n g$ $t o m a k e a p d f w i t h t h e s o l u t i o n .$ $A n y o n e i n t e r e s t e d i n t h e s o l u t i o n m a y$ $w r i t e m e a m a i l a t$ $a l e x_t i n k u t a r a_041103 @ a b v . b g$
	Commented by aleks041103 last updated on 31/Dec/21

	$A f t e r a l l t h e m a t h w e c o m e t o t h e a n s w e r :$ $\begin{matrix} a r e a = \frac{π}{2} (\frac{1}{2} + \underset{i = 1}{\sum^{\infty}} {s e c h}^{4} (l n (1 + \sqrt{2}) i)) \end{matrix}$
	Commented by aleks041103 last updated on 31/Dec/21

	$t u r n s o u t :$ $\frac{1}{r_{i}} = 1 + \frac{{(3 + 2 \sqrt{2})}^{i - 1} + {(3 - 2 \sqrt{2})}^{i - 1}}{2}$
	Commented by mr W last updated on 31/Dec/21

	$i k n e w y o u w i l l a r r i v e a t t h e f i n a l$ $s o l u t i o n . {i t}^{'} s p e r f e c t a g a i n s i r . y o u$ $h a v e s o l v e d a p r o b l e m w h i c h s e e m e d$ $u n s o l v a b l e . r e a l l y g r e a t j o b!$
	Commented by aleks041103 last updated on 31/Dec/21

	$m r W, I r e a l l y w a n t t o s h a r e m y s o l u t i o n$ $w i t h y o u . I f y o u w a n t y o u c a n t y p e y o u r e m a i l$ $a d d r e s s h e r e, o r j u s t w r i t e m e a n e m a i l$ $a t a l e x_t i n k u t a r a_041103 @ a b v . b g$
	Commented by mr W last updated on 31/Dec/21

	$t h a n k s a l o t s i r! i^{'} m v e r y i n t e r e s t e d .$ $i^{'} l l a l s o t r y i f i c o u l d c o m e t o t h e$ $s a m e r e s u l t . m a y b e y o u c a n p u b l i s h$ $y o u r s o l u t i o n h e r e s o t h a t m o r e$ $p e o p l e c a n b e n i f i t f r o m i t . y o u {d o n}^{'} t$ $n e e d t o t y p e w r i t e i t, y o u c a n a l s o$ $m a k e a p h o t o f r o m y o u r h a n d w r i t i n g$ $o r f r o m y o u r p d f, a n d t h e n p o s t t h e$ $i m a g e h e r e .$
	Commented by mr W last updated on 01/Jan/22

	$a l e x s i r :$ $i c o u l d c o m e t o t h e s a m e r e s u l t . p l e a s e$ $r e v i e w m y c o m p l e t e d w o r k i n g a b o v e .$ $t h a n k s f o r y o u r s u p p o r t!$
	Answered by aleks041103 last updated on 01/Jan/22

	Commented by aleks041103 last updated on 01/Jan/22

	Commented by aleks041103 last updated on 01/Jan/22

	Commented by aleks041103 last updated on 01/Jan/22

	Commented by aleks041103 last updated on 01/Jan/22

	Commented by aleks041103 last updated on 01/Jan/22

	Commented by aleks041103 last updated on 01/Jan/22

	Commented by aleks041103 last updated on 01/Jan/22

	Commented by aleks041103 last updated on 01/Jan/22

	Commented by aleks041103 last updated on 01/Jan/22

	Commented by aleks041103 last updated on 01/Jan/22

	Commented by aleks041103 last updated on 01/Jan/22

	Commented by aleks041103 last updated on 01/Jan/22

	Commented by aleks041103 last updated on 01/Jan/22

	Commented by mr W last updated on 01/Jan/22

	$i h a v e f o l l o w e d y o u r w o r k i n g . {i t}^{'} s$ $a m a z i n g h o w y o u a n a l y s e d t h e t h i n g$ $a n d g o t t h e r e c u r s i v e r e l a t i o n .$ $i u s e d a d i f f e r e n t w a y f o r m y s o l u t i o n,$ $w h i c h e v e n l o o k s a l i t t l e b i t e a s i e r$ $t h a n y o u r s . b u t t o b e h o n e s t, w i t h o u t$ $y o u r r e s u l t, i e v e n h a v e s t o p e d t o g o$ $f u r t h e r a n d h a v e a c c e p t e d t h a t {i t}^{'} s$ $i m p o s s i b l e t o s o l v e f o r r_{n} i n t e r m s$ $o f n .$
	Commented by mr W last updated on 01/Jan/22

	$w o w! w h a t a n e f f o r t y o u m a d e!$ $y o u h a v e r e a l l y d o n e a g r e a t j o b!$ $i t l o o k s l i k e a s c i e n t i f i c p a p e r w h i c h$ $y o u c a n p u b l i s h i n m a g a z i n e s .$
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