$\begin{array}{l}a) \quad A = \dfrac{1}{1.2.3} + \dfrac{1}{2.3.4} + \dots + \dfrac{1}{18.19.20} \\ \to 2A = \dfrac{2}{1.2.3} + \dfrac{2}{2.3.4} + \dots + \dfrac{2}{18.19.20} \\ \to 2A = \dfrac{3-1}{1.2.3} + \dfrac{4-2}{2.3.4} + \dots + \dfrac{20 -18}{18.19.20} \\ \to 2A = \dfrac{3}{1.2.3}-\dfrac{1}{1.2.3} + \dfrac{4}{2.3.4}-\dfrac{2}{2.3.4} + \dots + \dfrac{20}{18.19.20}-\dfrac{1}{18.19.20} \\ \to 2A = \dfrac{1}{1.2} - \dfrac{1}{2.3} + \dfrac{1}{2.3} - \dfrac{1}{3.4} + \dots + \dfrac{1}{18.19} - \dfrac{1}{19.20}\\ \to 2A = \dfrac{1}{2} - \dfrac{1}{19.20}\\ \to A = \dfrac{1}{4} - \dfrac{1}{19.40} < \dfrac{1}{4}\\ b) \quad B = \dfrac{36}{1.3.5} + \dfrac{36}{3.5.7} + \dots + \dfrac{36}{25.27.29}\\ \to B = 9\cdot\left(\dfrac{4}{1.3.5} + \dfrac{4}{3.5.7} + \dots + \dfrac{4}{25.27.29} \right)\\ \to B = 9\cdot\left(\dfrac{5 - 1}{1.3.5} + \dfrac{7 - 3}{3.5.7} + \dots + \dfrac{29 - 25}{25.27.29} \right)\\ \to B = 9\cdot\left(\dfrac{5}{1.3.5}-\dfrac{1}{1.3.5} + \dfrac{7}{3.5.7}-\dfrac{3}{3.5.7} + \dots + \dfrac{29}{25.27.29}-\dfrac{25}{25.27.29} \right)\\ \to B = 9\cdot\left(\dfrac{1}{1.3}-\dfrac{1}{3.5} + \dfrac{1}{3.5}-\dfrac{1}{5.7} + \dots + \dfrac{1}{25.27}-\dfrac{1}{27.29} \right)\\ \to B = 9\cdot\left(\dfrac{1}{3} - \dfrac{1}{27.29}\right)\\ \to B = 3 - \dfrac{9}{27.29} < 3 \end{array}$
