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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; (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
\(=\frac{4}{3}.\frac{9}{8}.\frac{16}{15}.\frac{25}{24}...\frac{100}{99}\)
\(1\frac{1}{3}\times1\frac{1}{8}\times1\frac{1}{15}\times1\frac{1}{24}\times...\times1\frac{1}{99}\)
\(=\frac{4}{3}\times\frac{9}{8}\times\frac{16}{15}\times\frac{25}{24}\times...\times\frac{100}{99}\)
\(=\frac{2\times2}{1\times3}\times\frac{3\times3}{2\times4}\times\frac{4\times4}{3\times5}\times\frac{5\times5}{4\times6}\times...\times\frac{10\times10}{9\times11}\)
\(=\frac{2}{1}\times\frac{10}{11}\)
\(=\frac{20}{11}\)
\(1\frac{1}{3}\)x \(1\frac{1}{8}\)x \(1\frac{1}{15}\)x .......x \(1\frac{1}{99}\)= \(\frac{4}{3}\)x \(\frac{9}{8}\)x\(\frac{16}{15}\)x..........x \(\frac{100}{99}\)
=\(\frac{2.2}{1.3}\)x\(\frac{3.3}{2.4}\)x\(\frac{4.4}{3.5}\)x.......x \(\frac{10.10}{9.11}\)= \(\frac{2.3.4....10}{1.2.3.....9}\)x \(\frac{2.3.4....10}{3.4.5....11}\)
= \(\frac{10}{1}\)x\(\frac{2}{11}\)=\(\frac{20}{11}\)
4/3 x 9/8 x16/15 x ..... x 100/99
=2x2/1x3 x 3x3/2x4 x 4x4/3x5 x ... x 10x10/9x11
=2x3x4x....x10/1x2x3x..x9 x 2x3x4x...x10/3x4x5x...x11
=10 x 2/11=20/11
\(1\frac{1}{3}.1\frac{1}{8}.1\frac{1}{15}.....1\frac{1}{99}\)
\(=\frac{4}{3}.\frac{9}{8}.\frac{16}{15}......\frac{100}{99}\)
\(=\frac{2.2}{1.3}.\frac{3.3}{2.4}.\frac{4.4}{3.5}.....\frac{10.10}{9.11}\)
\(=\frac{\left(2.3.4...10\right)\left(2.3.4....10\right)}{\left(1.2.3...9\right)\left(3.4.5....11\right)}\)
\(=\frac{10.2}{11}=\frac{20}{11}\)
\(71+52,5\times4=\frac{x+140}{x}+210\)
\(71+210=\frac{x+140}{x}+210\)
\(=>\frac{x+140}{x}=71\)
\(71=\frac{142}{2}\)\(\Rightarrow x=142-140=2\)
\(=\frac{4}{3}x\frac{9}{8}x\frac{16}{15}x\frac{25}{24}x...x\frac{100}{99}\)
\(=\frac{2x2x3x3x4x4x5x5x6x6x7x7x8x8x9x9x10x10}{1x3x2x4x3x5x4x6x5x7x6x8x7x9x8x10x9x11}=\frac{2x10}{11}=\frac{20}{11}\)
Lưu Hạ Băng
Ta có : \(1\frac{1}{3}x1\frac{1}{8}x1\frac{1}{15}x1\frac{1}{24}x...x1\frac{1}{99}\)
\(=\frac{4}{3}x\frac{9}{8}x\frac{16}{15}x\frac{25}{24}x...x\frac{100}{99}\)
\(=\frac{2x2}{1x3}x\frac{3x3}{2x4}x\frac{4x4}{3x5}x\frac{5x5}{4x6}x...x\frac{10x10}{9x11}\)
\(=\frac{2x2x3x3x4x4x5x5x...x10x10}{1x3x2x4x3x5x4x6x...x9x11}\)
\(=\frac{\left(2x3x4x5x...x10\right)x\left(2x3x4x5x...x10\right)}{\left(1x2x3x4x...x9\right)x\left(3x4x5x6x...x11\right)}\)
\(=\frac{10x2}{1x11}\)
\(=\frac{20}{11}\)
Chúc bn học tốt !!!
Kb vs mk heg =)) ^^