Bước 1: Tính $B$
Phân tích từ mẫu: $$\begin{aligned} \dfrac{99}{1}+\dfrac{98}{2}+\dfrac{97}{3}+....+ \dfrac{1}{99} & = \dfrac{100-1}{1}+ \dfrac{100-2}{2}+ \dfrac{100-3}{3}+...+ \dfrac{100-99}{99} \\ & = 100-99+ 100 \left( \dfrac{1}{2}+ \dfrac{1}{3}+ ...+ \dfrac{1}{99} \right) \\ & = 100 \left( \dfrac{1}{2}+ \dfrac{1}{3}+ ...+ \dfrac{1}{99}+ \dfrac{1}{100} \right) \end{aligned}$$
Vậy $B= \boxed{\dfrac{1}{100}}$
Bước 2: Tính $A$
Phân tích từ tử: $$\begin{aligned} 1+\dfrac{1}{3}+\dfrac{1}{5}+....+\dfrac{1}{97}+ \dfrac{1}{99} & = \left( 1+ \dfrac{1}{99} \right)+ \left( \dfrac{1}{3}+ \dfrac{1}{97} \right) +...+ \left( \dfrac{1}{49}+ \dfrac{1}{51} \right) + \dfrac{1}{50} \\ & = \dfrac{100}{1.99}+ \dfrac{100}{3.97}+...+ \dfrac{100}{49.51} \\ & = 50 \left( \dfrac{2}{1.99}+ \dfrac{2}{3.97}+...+ \dfrac{2}{49.51} \right) \\ & = 50 \left( \dfrac{1}{1.99}+\dfrac{1}{3.97}+\dfrac{1}{5.95}+.....+\dfrac{1}{97.3}+\dfrac{1}{99.1} \right) \end{aligned}$$
Vậy $A= \boxed{50}$.
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