[tex]\dfrac{x+1}{2}+\dfrac{2x+1}{3}+\dfrac{3x+1}{4}+...+\dfrac{99x+1}{100} = 99\\ \\ \dfrac{x+1}{2}+\dfrac{2x+1}{3}+\dfrac{3x+1}{4}+...+\dfrac{99x+1}{100} - 99 = 0 \\ \\ \Big(\dfrac{x+1}{2}-1\Big)+\Big(\dfrac{2x+1}{3}-1\Big)+\Big(\dfrac{3x+1}{4}-1\Big)+\\ ...+\Big(\dfrac{99x+1}{100}-1\Big) = 0 \\ \\ \dfrac{x+1-2}{2}+\dfrac{2x+1-3}{3}+\dfrac{3x+1-4}{4}+...+\dfrac{99x+1-100}{100}=0[/tex]
[tex]\dfrac{x-1}{2}+\dfrac{2(x-1)}{3}+\dfrac{3(x-1)}{4}+...+\dfrac{99(x-1)}{100} = 0 \\ \\ (x-1)\cdot \Big(\dfrac{1}{2}+\dfrac{2}{3}+\dfrac{3}{4}+...+\dfrac{99}{100}\Big) = 0 \Big|:\Big(\dfrac{1}{2}+\dfrac{2}{3}+\dfrac{3}{4}+...+\dfrac{99}{100}\Big)\\ \\ x-1 = 0 \\ \\ \Rightarrow \boxed{x = 1}[/tex]
