[tex]\dfrac{\frac{1}{100}+\frac{2}{101}+\frac{3}{102}+...+\frac{101}{200}-101}{\frac{1}{100}+\frac{1}{101}+\frac{1}{102}+...+\frac{1}{200}}=[/tex]
[tex]= \dfrac{(\frac{1}{100}-1)+(\frac{2}{101}-1)+(\frac{3}{102}-1)+...+(\frac{101}{200}-1)}{\frac{1}{100}+\frac{1}{101}+\frac{1}{102}+...+\frac{1}{200}}[/tex]
[tex]= \dfrac{\frac{1-100}{100}+\frac{2-101}{101}+\frac{3-102}{102}+...+\frac{101-200}{200}}{\frac{1}{100}+\frac{1}{101}+\frac{1}{102}+...+\frac{1}{200}}[/tex]
[tex]=\dfrac{\frac{-99}{100}+\frac{-99}{101}+\frac{-99}{102}+...+\frac{-99}{200}}{\frac{1}{100}+\frac{1}{101}+\frac{1}{102}+...+\frac{1}{200}}[/tex]
[tex]=\dfrac{-99\left(\frac{1}{100}+\frac{1}{101}+\frac{1}{102}+...+\frac{1}{200}\right)}{\frac{1}{100}+\frac{1}{101}+\frac{1}{102}+...+\frac{1}{200}}[/tex]
[tex]=-99\cdot \dfrac{\frac{1}{100}+\frac{1}{101}+\frac{1}{102}+...+\frac{1}{200}}{\frac{1}{100}+\frac{1}{101}+\frac{1}{102}+...+\frac{1}{200}}[/tex]
[tex]=-99\cdot 1[/tex]
[tex]=-99[/tex]
