Question

Tính: \(D=\dfrac{\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}+...+\dfrac{1}{2017}}{\dfrac{2011}{1}+\dfrac{2010}{2}+\dfrac{2009}{3}+...+\dfrac{1}{2011}}\)

Answer 1

\(D=\dfrac{\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}+...+\dfrac{1}{2012}}{\dfrac{2011}{1}+\dfrac{2010}{2}+...+\dfrac{1}{2011}}\)

Ta có mẫu của phân số trên là :

\(\dfrac{2011}{1}+\dfrac{2010}{2}+...+\dfrac{1}{2011}\)

\(=\left(\dfrac{2010}{2}+1\right)+\left(\dfrac{2009}{3}+1\right)+...+\left(\dfrac{1}{2011}+1\right)+1\)

=\(\dfrac{2012}{2}+\dfrac{2012}{3}+\dfrac{2012}{4}+...+\dfrac{2012}{2011}+\dfrac{2012}{2012}\)

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

Từ đó suy ra :

\(D=\dfrac{\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}+...+\dfrac{1}{2012}}{2012\left(\dfrac{1}{2}+\dfrac{1}{3}+...+\dfrac{1}{2012}\right)}=\dfrac{1}{2012}\)

Vậy \(D=\dfrac{1}{2012}\)

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