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Question Number 87379 by john santu last updated on 04/Apr/20
Π_(k = 2) ^(2010) ((k^2 −1)/k^2 ) = ?
$$\underset{\mathrm{k}\:=\:\mathrm{2}} {\overset{\mathrm{2010}} {\prod}}\:\frac{\mathrm{k}^{\mathrm{2}} −\mathrm{1}}{\mathrm{k}^{\mathrm{2}} }\:=\:? \\ $$
Commented by jagoll last updated on 04/Apr/20
Π_(k = 2) ^(2010) (1−(1/k^2 )) = (3/4)×(8/9)×((15)/(16))×((24)/(25))×...×((2011×2010)/(2010^2 ))
$$\underset{\mathrm{k}\:=\:\mathrm{2}} {\overset{\mathrm{2010}} {\prod}}\left(\mathrm{1}−\frac{\mathrm{1}}{\mathrm{k}^{\mathrm{2}} }\right)\:=\:\frac{\mathrm{3}}{\mathrm{4}}×\frac{\mathrm{8}}{\mathrm{9}}×\frac{\mathrm{15}}{\mathrm{16}}×\frac{\mathrm{24}}{\mathrm{25}}×…×\frac{\mathrm{2011}×\mathrm{2010}}{\mathrm{2010}^{\mathrm{2}} } \\ $$
Commented by john santu last updated on 04/Apr/20
= ((1.3)/2^2 ) × ((2.4)/3^2 ) ×((3.5)/4^2 )×((4.6)/5^2 )×((5.7)/6^2 )×...× ((2008.2010)/(2009^2 ))×((2009.2011)/(2010^2 )) = ((2×3^2 ×4^2 ×5^2 ×...2010×2011)/(2^2 ×3^2 ×4^2 ×...×2010^2 )) = ((2×2011)/(4×2010)) = ((2011)/(4020))
$$=\:\frac{\mathrm{1}.\mathrm{3}}{\mathrm{2}^{\mathrm{2}} }\:×\:\frac{\mathrm{2}.\mathrm{4}}{\mathrm{3}^{\mathrm{2}} }\:×\frac{\mathrm{3}.\mathrm{5}}{\mathrm{4}^{\mathrm{2}} }×\frac{\mathrm{4}.\mathrm{6}}{\mathrm{5}^{\mathrm{2}} }×\frac{\mathrm{5}.\mathrm{7}}{\mathrm{6}^{\mathrm{2}} }×…× \\ $$$$\frac{\mathrm{2008}.\mathrm{2010}}{\mathrm{2009}^{\mathrm{2}} }×\frac{\mathrm{2009}.\mathrm{2011}}{\mathrm{2010}^{\mathrm{2}} } \\ $$$$=\:\frac{\mathrm{2}×\mathrm{3}^{\mathrm{2}} ×\mathrm{4}^{\mathrm{2}} ×\mathrm{5}^{\mathrm{2}} ×…\mathrm{2010}×\mathrm{2011}}{\mathrm{2}^{\mathrm{2}} ×\mathrm{3}^{\mathrm{2}} ×\mathrm{4}^{\mathrm{2}} ×…×\mathrm{2010}^{\mathrm{2}} } \\ $$$$=\:\frac{\mathrm{2}×\mathrm{2011}}{\mathrm{4}×\mathrm{2010}}\:=\:\frac{\mathrm{2011}}{\mathrm{4020}} \\ $$
Answered by ajfour last updated on 04/Apr/20
P =Π_(k=2) ^(2010) (((k−1)(k+1))/(k×k)) = ((1/2)×(3/2))((2/3)×(4/3))((3/4)×(5/4))....(((2008)/(2009))×((2010)/(2009)))(((2009)/(2010))×((2011)/(2010))) = ((2009!×(((2011!)/(1×2))))/(2010!×2010!))=((2011)/(2010×2)) . = ((2011)/(4020)) .
$${P}\:=\underset{{k}=\mathrm{2}} {\overset{\mathrm{2010}} {\prod}}\frac{\left({k}−\mathrm{1}\right)\left({k}+\mathrm{1}\right)}{{k}×{k}} \\ $$$$\:\:=\:\left(\frac{\mathrm{1}}{\mathrm{2}}×\frac{\mathrm{3}}{\mathrm{2}}\right)\left(\frac{\mathrm{2}}{\mathrm{3}}×\frac{\mathrm{4}}{\mathrm{3}}\right)\left(\frac{\mathrm{3}}{\mathrm{4}}×\frac{\mathrm{5}}{\mathrm{4}}\right)….\left(\frac{\mathrm{2008}}{\mathrm{2009}}×\frac{\mathrm{2010}}{\mathrm{2009}}\right)\left(\frac{\mathrm{2009}}{\mathrm{2010}}×\frac{\mathrm{2011}}{\mathrm{2010}}\right) \\ $$$$\:\:=\:\frac{\mathrm{2009}!×\left(\frac{\mathrm{2011}!}{\mathrm{1}×\mathrm{2}}\right)}{\mathrm{2010}!×\mathrm{2010}!}=\frac{\mathrm{2011}}{\mathrm{2010}×\mathrm{2}}\:. \\ $$$$\:\:=\:\frac{\mathrm{2011}}{\mathrm{4020}}\:. \\ $$

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