$B=\frac{1}{1+3}+\frac{1}{1+3+5}+\frac{1}{1+3+5+7}+....+\frac{1}{1+3+5+7+...+101}$
$B=\frac{\frac{1}{(1+3).2}}{2}+\frac{\frac{1}{(1+5).3}}{2}+....+\frac{\frac{1}{(1+101).51}}{2}$
$B=\frac{2}{4.2}+\frac{2}{6.3}+...+\frac{2}{102.51}$
$B=\frac{1}{2.2}+\frac{1}{3.3}+...+\frac{1}{51.51}$
$⇒ B < \frac{1}{2.2}+(\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{51.51})$
$⇒ B < \frac{1}{4}+(\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{50}-\frac{1}{51})$
$⇒ B < \frac{1}{4}+(\frac{1}{2}-\frac{1}{51})$
$⇒ A < \frac{1}{4}+\frac{1}{2}=\frac{3}{4}$