Question

1. so sanh a với b biet

\(a=\frac{1x2}{2x2}x\frac{2x3}{3x3}x\frac{3x4}{4x4}x\frac{4x5}{5x5}x....x\frac{2012x2013}{2013x2013}\)

\(b=\frac{2012x2013-2012x2012}{2012x2011+2012x2}\)

 

Answer 1

1.

\(A=\frac{1.2}{2.2}.\frac{2.3}{3.3}.\frac{3.4}{4.4}......\frac{2012.2013}{2013.2013}\)

\(A=\frac{1}{2}.\frac{2}{3}.\frac{3}{4}.........\frac{2012}{2013}\)

\(A=\frac{1.2.3.4.....2012}{2.3.4.5......2013}\)

\(A=\frac{1}{2013}\)

 

\(B=\frac{2012.2013-2012.2012}{2012.2011+2012.2}\)

\(B=\frac{2012\left(2013-2012\right)}{2012\left(2011+2\right)}\)

\(B=\frac{2013-2012}{2011+2}\)

\(B=\frac{1}{2013}\)

\(Vì:\frac{ 1}{2013}=\frac{1}{2013}\)

\(\Rightarrow\frac{1.2}{2.2}.\frac{2.3}{3.3}.\frac{3.4}{4.4}......\frac{2012.2013}{2013.2013}=\frac{2012.2013-2012.2012}{2012.2011+2012.2}\)

\(Hay: A=B\)

Answer 2

\(A=\frac{1\times2}{2\times2}\times\frac{2\times3}{3\times3}\times\frac{3\times4}{4\times4}\times\frac{4\times5}{5\times5}\times...\times\frac{2012\times2013}{2013\times2013}\)

\(\Rightarrow A=\frac{1}{2}\times\frac{2}{3}\times\frac{3}{4}\times\frac{4}{5}\times...\times\frac{2012}{2013}\)

\(\Rightarrow A=\frac{1\times2\times3\times4\times...\times2012}{2\times3\times4\times5\times...\times2013}\)

\(\Rightarrow A=\frac{1}{2013}\)

\(B=\frac{2012\times2013-2012\times2012}{2012\times2011+2012\times2}\)

\(\Rightarrow B=\frac{2012\times\left(2013-2012\right)}{2012\times\left(2011+2\right)}\)

\(\Rightarrow B=\frac{2012\times1}{2012\times2013}\)

\(\Rightarrow B=\frac{1}{2013}\)