a) A = \(\frac{5}{1.4}+\frac{29}{4.7}+\frac{71}{7.10}+....+\frac{10301}{100.103}\) (có 34 số hạng)
A = \(\frac{4+1}{1.4}+\frac{4.7+1}{4.7}+\frac{7.10+1}{7.10}+....+\frac{100.103+1}{103.100}\)
A = \(1+\frac{1}{1.4}+1+\frac{1}{4.7}+1+\frac{1}{7.10}+....+1+\frac{1}{100.103}\)
A = \(1.34+\frac{1}{3}.\left(\frac{3}{1.4}+\frac{3}{4.7}+\frac{3}{7.10}+...+\frac{3}{100.103}\right)\)
A = \(34+\frac{1}{3}.\left(1-\frac{1}{4}+\frac{1}{4}-\frac{1}{7}+\frac{1}{7}-\frac{1}{10}+...+\frac{1}{100}-\frac{1}{103}\right)\)
A = \(34+\frac{1}{3}.\left(1-\frac{1}{103}\right)\)
A = \(34+\frac{1}{3}\cdot\frac{102}{103}\)
A = \(34+\frac{34}{103}=\frac{3536}{103}\)
\(Q=\frac{1.3}{3.5}+\frac{2.4}{5.7}+\frac{3.5}{7.9}+...+\frac{49.51}{99.101}\)
\(\Rightarrow Q=\frac{\left(2-1\right)\left(2+1\right)}{\left(2.2-1\right)\left(2.2+1\right)}+\frac{\left(3-1\right)\left(3+1\right)}{\left(3.2-1\right)\left(3.2+1\right)}+...+\frac{\left(50-1\right)\left(50+1\right)}{\left(50.2-1\right)\left(50.2+1\right)}\)
\(\Rightarrow Q=\frac{1}{4}-\frac{3}{8}\left(\frac{1}{2.2-1}-\frac{1}{2.2+1}\right)+\frac{1}{4}-\frac{3}{8}\left(\frac{1}{3.2-1}-\frac{1}{3.2+1}\right)+...+\)\(\frac{1}{4}-\frac{3}{8}\left(\frac{1}{50.2-1}-\frac{1}{50.2+1}\right)\)
\(\Rightarrow Q=49.\frac{1}{4}-49.\frac{3}{8}\left(\frac{1}{2.2-1}-\frac{1}{2.2+1}+...+\frac{1}{50.2-1}-\frac{1}{50.2+1}\right)\)
\(\Rightarrow Q=\frac{49}{4}-\frac{147}{8}\left(\frac{1}{3}-\frac{1}{5}+\frac{1}{5}-\frac{1}{7}+\frac{1}{7}-\frac{1}{9}+...+\frac{1}{99}-\frac{1}{101}\right)\)
\(\Rightarrow Q=\frac{49}{4}-\frac{147}{8}\left(\frac{1}{3}-\frac{1}{101}\right)\)
\(\Rightarrow Q=\frac{49}{4}-\frac{147}{8}.\frac{98}{303}\)
\(\Rightarrow Q=\frac{49}{4}-\frac{2401}{404}\)
\(\Rightarrow Q=\frac{637}{101}\)
