\(\frac{1}{1.2.3}+\frac{1}{2.3.4}+...+\frac{1}{2004.2005.2006}\)

\(=\frac{1}{2}\left(\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+...+\frac{1}{2004.2005}-\frac{1}{2005.2006}\right)\)

\(=\frac{1}{2}\left(\frac{1}{2}-\frac{1}{2005.2006}\right)\)

\(=\frac{1}{4}-\frac{1}{2.2005.2006}\)

cách làm như sau

\(C=\frac{2}{2}.\left[\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+....+\frac{1}{98.99}-\frac{1}{99.100}\right]\)

\(C=1\left[\frac{1}{2}-\frac{1}{9900}\right]\)

\(C=\frac{4949}{9900}\)

cần làm ra ko

CÓ CHỨ

1/2-1/2010.2011

cậu có biết tách ko?

nếu cậu biết tách ra thành cách hiệu thì sẽ làm được nhanh thôi

A = \(\frac{2}{1.2.3}+\frac{2}{2.3.4}+....+\frac{2}{37.38.39}\)

A = \(\frac{3-1}{1.2.3}+\frac{4-2}{2.3.4}+...+\frac{39-37}{37.38.39}\)

A = \(\frac{3}{1.2.3}-\frac{1}{1.2.3}+\frac{4}{2.3.4}-\frac{2}{2.3.4}+....+\frac{39}{37.38.39}-\frac{37}{37.38.39}\)

A = \(\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+....+\frac{1}{37.18}-\frac{1}{38.39}\)

A = \(\frac{1}{2}-\frac{1}{38.39}\)

A = \(\frac{370}{741}\)

\(\left(\frac{2}{1.2.3}+\frac{2}{2.3.4}+...+\frac{2}{19.20.21}\right).x=5\)

\(\left(\frac{3-1}{1.2.3}+\frac{4-2}{2.3.4}+...+\frac{21-19}{19.20.21}\right).x=5\)

 \(\left(\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+...+\frac{1}{19.20}-\frac{1}{20.21}\right).x=5\)

 \(\left(\frac{1}{1.2}-\frac{1}{20.21}\right).x=5\)

 \(\frac{209}{420}.x=5\)

\(\Rightarrow x=5\div\frac{209}{420}=\frac{2100}{209}\)

\(\left(\frac{2}{1.2.3}+\frac{2}{2.3.4}+...+\frac{2}{19.20.21}\right).x=5\)

\(\left(\frac{1}{1.2.3}+\frac{1}{2.3.4}+...+\frac{1}{19.20.21}\right).2.x=5\)

\(\left(\frac{1}{2}.\left(\frac{1}{1.2}-\frac{1}{2.3}\right)+\frac{1}{2}.\left(\frac{1}{2.3}-\frac{1}{3.4}\right)+...+\frac{1}{2}.\left(\frac{1}{19.20}-\frac{1}{20.21}\right)\right).x.2=5\)

\(\left(\frac{1}{2}.\left(\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+...+\frac{1}{19.20}-\frac{1}{20.21}\right)\right).x=5\div2\)

\(\left(\frac{1}{2}.\left(\frac{1}{1.2}-\frac{1}{20.21}\right)\right).x=2,5\)

\(\left(\frac{1}{2}.\left(\frac{1}{2}-\frac{1}{420}\right)\right).x=2,5\)

\(\left(\frac{1}{2}\times\frac{209}{420}\right)\times x=2,5\)

\(\frac{209}{840}\times x=2,5\)

\(x=2,5\div\frac{209}{840}=10\frac{10}{209}\)

S=1/1.2 - 1/2.3 + 1/2.3 - 1/3.4 +...+ 1/2010.2011 - 1/2011.2012

S=1/1.2 - 1/2011.2012<1/2

=>S<P

75:x=3(du 3 )

\(S=\frac{2}{1.2.3}+\frac{2}{2.3.4}+\frac{2}{3.4.5}+........+\frac{2}{2010.2011.2012}\)

\(=\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+\frac{1}{3.4}-\frac{1}{4.5}+......+\frac{1}{2010.2011}-\frac{1}{2011.2012}\)

\(=\frac{1}{1.2}-\frac{1}{2011.2012}\)

\(=\frac{1}{2}-\frac{1}{2011.2012}\)

mà \(\frac{1}{2}-\frac{1}{2011.2012}< \frac{1}{2}\)

\(\Rightarrow S< P\)

Giúp mình

Ta có: \(\frac{1}{1.2.3}+\frac{1}{2.3.4}+...+\frac{1}{2004.2005.2006}\)

\(=\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+...+\frac{1}{2004.2005}-\frac{1}{2005.2006}\)

\(=\frac{1}{1.2}-\frac{1}{2005.2006}\)

\(=\frac{1}{2}-\frac{1}{4022030}\)

\(=-40220295.\)

\(=1-\frac{1}{2}-\frac{1}{3}+\frac{1}{2}-\frac{1}{3}-\frac{1}{4}+...-\frac{1}{2005}+\frac{1}{2004}-\frac{1}{2005}-\frac{1}{2006}\)

\(=1-\frac{1}{2006}=\frac{2005}{2006}\)

s=1/1*2-1/2*3+1/2*3-1/3*4+....+1/2009*2010-1/210*2011

 =1/1*2-1/2010*2011

<1/1*2

\(S=\frac{2}{1\cdot2\cdot3}+\frac{2}{2\cdot3\cdot4}+\frac{2}{3\cdot4\cdot5}+...+\frac{2}{2009\cdot2010\cdot2011}\)

\(S=\frac{1}{1\cdot2}-\frac{1}{2\cdot3}+\frac{1}{2\cdot3}-\frac{1}{3\cdot4}+\frac{1}{3\cdot4}-\frac{1}{4\cdot5}+...+\frac{1}{2009\cdot2010}-\frac{1}{2010\cdot2011}\)

\(S=\frac{1}{1\cdot2}-\frac{1}{2010\cdot2011}\)

\(S=\frac{1}{2}-\frac{1}{2010\cdot2011}< \frac{1}{2}\)

=> S < P