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\(S=\dfrac{2}{1.2.3}+\dfrac{2}{2.3.4}+\dfrac{2}{3.4.5}+...+\dfrac{2}{2009.2010.2011}\)
\(=\dfrac{1}{1.2}-\dfrac{1}{2.3}+\dfrac{1}{2.3}-\dfrac{1}{3.4}+\dfrac{1}{3.4}-\dfrac{1}{4.5}+...+\dfrac{1}{2009.2010}-\dfrac{1}{2010.2011}\)
\(=\dfrac{1}{1.2}-\dfrac{1}{2010.2011}\)
\(=\dfrac{1}{2}-\dfrac{1}{4042110}< \dfrac{1}{2}\)
\(\Rightarrow\) \(S< P\)
Vậy \(S< P\)
\(A=\dfrac{1}{1.2.3}+\dfrac{1}{2.3.4}+\dfrac{1}{3.4.5}+...+\dfrac{1}{2014.2015.2016}\)
\(A=\dfrac{1}{2}\left(\dfrac{1}{1.2}+\dfrac{1}{2.3}+\dfrac{1}{2.3}+\dfrac{1}{3.4}+...+\dfrac{1}{2014.2015}+\dfrac{1}{2015.2016}\right)\)
\(A=\dfrac{1}{2}\left(\dfrac{1}{1.2}-\dfrac{1}{2015.2016}\right)=\dfrac{1}{2}\left(\dfrac{1}{2}-\dfrac{1}{2015.2016}\right)\)
\(A=\dfrac{1}{4}-\dfrac{1}{2.2015.2016}< \dfrac{1}{4}\)
\(=>A< \dfrac{1}{4}\)
Chúc bn học tốt
Ta có :
\(S=\dfrac{1}{1.2.3}+\dfrac{1}{2.3.4}+..............+\dfrac{1}{98.99.100}\)
\(S=\dfrac{1}{2}\left(\dfrac{2}{1.2.3}+\dfrac{2}{2.3.4}+................+\dfrac{2}{98.99.100}\right)\)
\(S=\dfrac{1}{2}\left(\dfrac{1}{1.2}-\dfrac{1}{2.3}+\dfrac{1}{2.3}-\dfrac{1}{3.4}+...........+\dfrac{1}{98.99}-\dfrac{1}{99.100}\right)\)
\(S=\dfrac{1}{2}\left(\dfrac{1}{1.2}-\dfrac{1}{99.100}\right)\)
\(S=\dfrac{1}{2}\left(\dfrac{1}{2}-\dfrac{1}{9900}\right)\)
\(S=\dfrac{1}{2}.\dfrac{4949}{9900}\)
\(S=\dfrac{4949}{19800}\)
~ Chúc bn học tốt ~