Đặt\(B=\)\(\dfrac{5^2}{1.6}+\dfrac{5^2}{6.11}+...+\dfrac{5^2}{26.31}\)
\(=5^2.\left(\dfrac{1}{1.6}+\dfrac{1}{6.11}+...+\dfrac{1}{26.31}\right)\)
\(=25.\left(\dfrac{1}{1.6}+\dfrac{1}{6.11}+...+\dfrac{1}{26.31}\right)\)
Đặt \(A=\dfrac{1}{1.6}+\dfrac{1}{6.11}+...+\dfrac{1}{26.31}\)
\(\Rightarrow5A=\dfrac{5}{1.6}+\dfrac{5}{6.11}+...+\dfrac{5}{26.31}\)
\(\Rightarrow5A=\dfrac{6-1}{1.6}+\dfrac{11-6}{6.11}+...+\dfrac{31-26}{26.31}\)
\(\Rightarrow5A=\dfrac{6}{1.6}-\dfrac{1}{1.6}+\dfrac{11}{6.11}-\dfrac{6}{6.11}+...+\dfrac{31}{26.31}-\dfrac{26}{26.31}\)
\(\Rightarrow5A=\dfrac{1}{1}-\dfrac{1}{6}+\dfrac{1}{6}-\dfrac{1}{11}+...+\dfrac{1}{26}-\dfrac{1}{31}\)
\(\Rightarrow5A=\dfrac{1}{1}-\dfrac{1}{31}\)
\(\Rightarrow5A=\dfrac{31}{31}-\dfrac{1}{31}\)
\(\Rightarrow5A=\dfrac{30}{31}\)
\(\Rightarrow A=\dfrac{30}{31}:5=\dfrac{30}{31}.\dfrac{1}{5}=\dfrac{6}{31}\)
\(\Rightarrow B=25.\dfrac{6}{31}=\) \(\dfrac{150}{31}\) \(=4\dfrac{26}{31}\)
Vậy \(\dfrac{5^2}{1.6}+\dfrac{5^2}{6.11}+...+\dfrac{5^2}{26.31}=4\dfrac{26}{31}\)
