Question

 so sánh \(A=\frac{1}{1.2.3}+\frac{1}{2.3.4}+\frac{1}{3.4.5}+...+\frac{1}{2012.2013.2014}\) với \(\frac{1}{4}\)

Answer 1

A<\(\frac{1}{4}\)

Answer 2

A>1/4

Answer 3

A<1/4 BN NHÉ

Answer 4

\(A=\frac{1}{1.2.3}+\frac{1}{2.3.4}+\frac{1}{3.4.5}+...+\frac{1}{2012.2013.2014}\)

\(A=\frac{3-1}{1.2.3}+\frac{4-2}{2.3.4}+\frac{5-3}{3.4.5}+...+\frac{2014-2012}{2012.2013.2014}\)

\(A=\frac{1}{2}.\left(\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+\frac{1}{3.4}-\frac{1}{4.5}+...+\frac{1}{2012.2013}-\frac{1}{2013.2014}\right)\)

\(A=\frac{1}{2}.\left(\frac{1}{1.2}-\frac{1}{2013.2014}\right)=\frac{1}{2}.\frac{2027090}{4054182}\)

\(A=\frac{2027090}{8108364}\)

Ta thấy \(\frac{2027090}{8108364}=0,2499.....\)  \(\frac{1}{4}=0,25\)

\(\Rightarrow A< \frac{1}{4}\)