Question

B=\(\left(\dfrac{1}{2}+\dfrac{1}{3}+...+\dfrac{1}{2013}\right):\left(2012+\dfrac{2011}{2}+...+\dfrac{2}{2011}+\dfrac{1}{2012}\right)\).Tính B

Answer 1

Đặt \(B=A\div C\)

\(C=2012+\dfrac{2011}{2}+...+\dfrac{1}{2012}=2012+\dfrac{2013-2}{2}+\dfrac{2013-3}{3}+...+\dfrac{2013-2012}{2012}\)

\(C=2012+\dfrac{2013}{2}+\dfrac{2013}{3}+...+\dfrac{2013}{2012}-1-1-...-1\)

\(C=2012+2013\left(\dfrac{1}{2}+\dfrac{1}{3}+...+\dfrac{1}{2012}\right)-2011\)

\(C=1+2013\left(\dfrac{1}{2}+\dfrac{1}{3}+...+\dfrac{1}{2012}\right)=\dfrac{2013}{2013}+2013\left(\dfrac{1}{2}+\dfrac{1}{3}+...+\dfrac{1}{2012}\right)\)

\(C=2013\left(\dfrac{1}{2}+\dfrac{1}{3}+...+\dfrac{1}{2013}\right)=2013.A\)

\(\Rightarrow B=\dfrac{A}{C}=\dfrac{1}{2013}\)