prove that (1/2)×(3/4)×(5/6)×(7/8)×(9/(10))...×((99)/(100))>(1/(13))


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Question Number 193760 by York12 last updated on 19/Jun/23

$p r o v e t h a t$ $\frac{1}{2} \times \frac{3}{4} \times \frac{5}{6} \times \frac{7}{8} \times \frac{9}{10} . . . \times \frac{99}{100} > \frac{1}{13}$
	Answered by MM42 last updated on 19/Jun/23

	$c a n b r s h o w : \underset{i = 1}{\prod^{n}} \frac{2 i - 1}{2 i} > \frac{11}{10 \sqrt{4 n + 1}}$ $\Rightarrow f o r n = 50 t o h a v e$ $\frac{1}{2} \times \frac{3}{4} \times . . . \times \frac{99}{100} > \frac{11}{10 \sqrt{201}} > \frac{1}{13}$
	Commented by York12 last updated on 19/Jun/23

	$s i r w h e r e c a n I l e a r n t h a t$
	Commented by York12 last updated on 19/Jun/23

	$\frac{11}{10 \sqrt{4 n + 1}} h o w c o u l d y o u k n o w t h a t$ $I f e e l l i k e t h e r e i s s o m e t h i n g I d o n o t k n o w$
	Commented by MM42 last updated on 19/Jun/23

	$s . i \to i = 1 \Rightarrow \frac{1}{2} > \frac{11}{10 \sqrt{5}} \Rightarrow 125 > 121 ✓$ $i . h \to i = k \Rightarrow \underset{i = 1}{\prod^{k}} \frac{2 i - 1}{2 i} > \frac{11}{10 \sqrt{4 k + 1}}$ $i . h \to i = k + 1 \Rightarrow \underset{i = 1}{\prod^{k + 1}} \frac{2 i - 1}{2 i} > \frac{11}{10 \sqrt{4 k + 5}}$ $\underset{i = 1}{\prod^{k + 1}} \frac{2 i - 1}{2 i} = \underset{i = 1}{\prod^{k}} \frac{2 i - 1}{2 i} \times \frac{2 k + 1}{2 k + 2} >$ $\frac{11}{10 \sqrt{4 k + 1}} \times \frac{2 k + 1}{2 k + 2} \overset{★}{>} \frac{11}{10 \sqrt{4 k + 5}}$ $★ \to (2 k + 1) \sqrt{4 k + 5} > (2 k + 2) \sqrt{4 k + 1}$ $(4 k^{2} + 4 k + 1) (4 k + 5) > 4 (k^{2} + 2 k + 1) (4 k + 1)$ $16 k^{3} + 36 k^{2} + 24 k + 5 > 16 k^{3} + 36 k^{2} + 24 k + 4$ $5 > 1 ✓ i t a l w a y s e x i s t s$
	Commented by York12 last updated on 20/Jun/23

	$t h a t i s p r o o f b y i n d u c t i o n s$ $b u t s i r c a n y o u r e c o m m e n d m e a b o o k s t o u s e$ $, I a m p r e p a r i n g f o r I M O$
	Commented by MM42 last updated on 20/Jun/23

	$H e l l o . d e a r f r i e n d . i a m a r e t i r e d$ $t e a c h e r f r o m e i r a n . t h e b o o k s a v a i l a b l e t o$ $m e a r e w h i t h p e r s i a n t r a n s l a t i o n .$ $b u t i w i l l s e n d y o u t h e n a m e s o f s o m e b o o k s .$ $i h o o p r i t u s e f u l f o r y o u$ $g o o d l u c k .$
	Commented by MM42 last updated on 20/Jun/23

	Commented by MM42 last updated on 20/Jun/23

	Commented by MM42 last updated on 20/Jun/23

	Commented by talminator2856792 last updated on 21/Jun/23

	$entertaining mathematical puzzles, martin gardner$
	Commented by York12 last updated on 23/Jul/23

	$t h a n k s s o m u c h s i r$
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