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=13/12x14/13x15/14x16/15x...x2006/2005x2007/2006x2008/2007
=2008/12
=502/3
A = 1\(\dfrac{1}{12}\) \(\times\) 1\(\dfrac{1}{13}\) \(\times\) 1\(\dfrac{1}{14}\) \(\times\) 1\(\dfrac{1}{15}\) \(\times\) ... \(\times\) 1\(\dfrac{1}{2005}\) \(\times\) 1\(\dfrac{1}{2006}\) \(\times\) 1\(\dfrac{1}{2007}\)
A = ( 1 + \(\dfrac{1}{12}\)) \(\times\) ( 1 + \(\dfrac{1}{13}\)) \(\times\) ( 1 + \(\dfrac{1}{14}\)) \(\times\)...\(\times\) ( 1 + \(\dfrac{1}{2006}\))\(\times\)(1+\(\dfrac{1}{2007}\))
A = \(\dfrac{13}{12}\) \(\times\) \(\dfrac{14}{13}\) \(\times\) \(\dfrac{15}{14}\) \(\times\) ...\(\times\) \(\dfrac{2007}{2006}\) \(\times\) \(\dfrac{2008}{2007}\)
A = \(\dfrac{13\times14\times15\times...\times2007}{13\times14\times15\times...\times2007}\) \(\times\) \(\dfrac{2008}{12}\)
A = 1 \(\times\) \(\dfrac{502}{3}\)
A = \(\dfrac{502}{3}\)
Ta có công thức tổng quát:
\(\dfrac{k}{n\cdot\left(n+k\right)}=\dfrac{1}{n}-\dfrac{1}{n+k}\)
\(a,A=\dfrac{1}{5\cdot8}+\dfrac{1}{8\cdot11}+...+\dfrac{1}{x\left(x+3\right)}\\ =\dfrac{1}{3}\left(\dfrac{3}{5\cdot8}+\dfrac{3}{8\cdot11}+...+\dfrac{3}{x\left(x+3\right)}\right)\\ =\dfrac{1}{3}\left(\dfrac{1}{5}-\dfrac{1}{8}+\dfrac{1}{8}-\dfrac{1}{11}+...+\dfrac{1}{x}-\dfrac{1}{x+3}\right)\\ =\dfrac{1}{3}\cdot\left(\dfrac{1}{5}-\dfrac{1}{x+3}\right)\\ =\dfrac{1}{3}\cdot\dfrac{x-2}{5\left(x+3\right)}\\ =\dfrac{x-2}{15\left(x+3\right)}\)
Theo đề bài ta có:
\(A=\dfrac{101}{1540}\\ \Rightarrow\dfrac{x-2}{15\left(x+3\right)}=\dfrac{101}{1540}\\ \Rightarrow\dfrac{x-2}{x+3}=\dfrac{303}{308}\\ \Rightarrow\dfrac{x-2}{x+3}=\dfrac{305-2}{305+3}\\ \Rightarrow x=305\)
a) \(\dfrac{2}{3}+\dfrac{3}{5}=\dfrac{10}{15}+\dfrac{9}{15}=\dfrac{19}{15}\)
a) \(\dfrac{7}{12}-\dfrac{2}{7}+\dfrac{1}{12}=\dfrac{2}{3}-\dfrac{2}{7}=\dfrac{14}{21}-\dfrac{6}{21}=\dfrac{8}{21}\)
a; (5142 - 17 x 8 + 242 : 11) x (27 - 3 x 9)
= (5142 - 17 x 8 + 242 : 11) x (27 - 27)
= (5142 - 17 x 8 + 242 : 11) x 0
= 0
b;
(1 + \(\dfrac{1}{2}\)) \(\times\) (1 + \(\dfrac{1}{3}\)) \(\times\) ( 1 + \(\dfrac{1}{4}\)) \(\times\) ... \(\times\) (1 + \(\dfrac{1}{2010}\)) \(\times\)(1 + \(\dfrac{1}{2011}\))
= \(\dfrac{2+1}{2}\) \(\times\) \(\dfrac{3+1}{3}\) \(\times\) \(\dfrac{4+1}{4}\)\(\times\) ... \(\times\) \(\dfrac{2010+1}{2010}\)\(\times\) \(\dfrac{2011+1}{2011}\)
= \(\dfrac{3}{2}\)\(\times\)\(\dfrac{4}{3}\)\(\times\)\(\dfrac{5}{4}\)\(\times\)...\(\times\)\(\dfrac{2011}{2010}\)\(\times\)\(\dfrac{2012}{2011}\)
= \(\dfrac{2012}{2}\)
= 1006
\(\dfrac{13+x}{20}\) = \(\dfrac{3}{4}\)
13 + \(x\) = 20 \(\times\) \(\dfrac{3}{4}\)
13 + \(x\) = 15
\(x\) = 15 - 13
\(x\) = 2
Cách khác :
\(\dfrac{13+x}{20}=\dfrac{3}{4}\)
\(\dfrac{13+x}{20}=\dfrac{15}{20}\)
\(13+x=15\)
\(x=15-13\)
\(x=2\)
a; A = \(\dfrac{4026\times2014+4030}{2013\times2016-2011}\)
A = \(\dfrac{2\times\left(2013\times2014+2015\right)}{2013\times2016-2011}\)
A = \(\dfrac{2\times\left(2013\times2016-2013\times2+2015\right)}{2013\times2016-2011}\)
A = \(\dfrac{2\times\left(2013\times2016-4026+2015\right)}{2013\times2016-2011}\)
A = \(\dfrac{2\times\left(2013\times2016-2011\right)}{2013\times2016-2011}\)
A = 2
\(71+52,5\times4=\frac{x+140}{x}+210\)
\(71+210=\frac{x+140}{x}+210\)
\(=>\frac{x+140}{x}=71\)
\(71=\frac{142}{2}\)\(\Rightarrow x=142-140=2\)
`x+3/4=4/3`
`x=4/3-3/4`
`x=16/12 - 9/12`
`x=7/12`
________________
`1/8:x=1/2`
`x=1/8:1/2`
`x=1/8 xx 2`
`x=2/8`
`x=1/4`
______________________
`x-3/11=9/22`
`x=9/22+3/11`
`x=9/22 + 6/22`
`x=15/22`
__________________
`x+1/4 xx 6/7=3/5`
`x+3/14 = 3/5`
`x=3/5-3/14`
`x=27/70`
__________________
`(2/5+4/7):x=17/5`
`(14/35 + 20/35):x=17/5`
`34/35 : x=17/5`
`x=34/35:17/5`
`x=34/35 xx 5/17`
`x=2/7`
_____________________
`7/3-x=11/5:5/6`
`7/3-x= 11/5 xx 6/5`
`7/3-x=66/25`
`x=7/3 - 66/25`
`x=-23/75`
a) \(x\) + \(\dfrac{3}{4}\) = \(\dfrac{4}{3}\) b) \(\dfrac{1}{8}\) : \(x\) = \(\dfrac{1}{2}\)
\(x\) = \(\dfrac{4}{3}\) - \(\dfrac{3}{4}\) \(x\) = \(\dfrac{1}{8}\) : \(\dfrac{1}{2}\)
\(x\) = \(\dfrac{7}{12}\) \(x\) = \(\dfrac{1}{4}\)
c) \(x\) + \(\dfrac{1}{4}\) x \(\dfrac{6}{7}\) = \(\dfrac{3}{5}\) d) \(x\) + \(\dfrac{1}{4}\) x \(\dfrac{6}{7}\) = \(\dfrac{3}{5}\)
\(x\) + \(\dfrac{3}{14}\) = \(\dfrac{3}{5}\) \(x\) + \(\dfrac{3}{14}\) = \(\dfrac{3}{5}\)
\(x\) = \(\dfrac{3}{5}\) - \(\dfrac{3}{14}\) \(x\) = \(\dfrac{3}{5}\) - \(\dfrac{3}{14}\)
\(x\) = \(\dfrac{27}{70}\) \(x\) = \(\dfrac{27}{70}\)
e) \(\left(\dfrac{2}{5}+\dfrac{4}{7}\right)\) : \(x\) = \(\dfrac{17}{5}\) f) \(\dfrac{7}{3}\) - \(x\) = \(\dfrac{11}{5}\) : \(\dfrac{5}{6}\)
\(\dfrac{34}{35}\) : \(x\) = \(\dfrac{17}{5}\) \(\dfrac{7}{3}\) - \(x\) = \(\dfrac{66}{25}\)
\(x\) = \(\dfrac{34}{35}\) : \(\dfrac{17}{5}\) \(x\) = \(\dfrac{7}{3}\) - \(\dfrac{66}{25}\)
\(x\) = \(\dfrac{2}{7}\) \(x\) = - \(\dfrac{23}{75}\)