Đặt: \(A=\frac{1}{99.97}-\frac{1}{97.95}-\frac{1}{95.93}-...-\frac{1}{5.3}-\frac{1}{3.1}\)
\(=\frac{1}{99.97}-\left(\frac{1}{97.95}+\frac{1}{95.93}+...+\frac{1}{3.1}\right)\)
\(=\frac{1}{2}\left(\frac{1}{97}-\frac{1}{99}\right)-\frac{1}{2}\left(1-\frac{1}{3}+...+\frac{1}{93}-\frac{1}{95}+\frac{1}{95}-\frac{1}{97}\right)\)
\(=\frac{1}{2}\left(\frac{1}{97}-\frac{1}{99}\right)-\frac{1}{2}\left(1-\frac{1}{97}\right)\)
\(=\frac{1}{2}.\frac{1}{97}-\frac{1}{2}.\frac{1}{99}-\frac{1}{2}+\frac{1}{2}.\frac{1}{97}\)
\(=-\frac{4751}{9603}\)
Vậy ....
\(\frac{1}{99.97}-\frac{1}{97.95}-...-\frac{1}{5.3}-\frac{1}{3.1}\)
\(=\frac{1}{99.97}-\left(\frac{1}{97.95}+...+\frac{1}{5.3}+\frac{1}{3.1}\right)\)
\(=\frac{1}{99.97}-\left(\frac{1}{1.3}+\frac{1}{3.5}+...+\frac{1}{95.97}\right)\left(1\right).\)
Đặt \(A=\frac{1}{1.3}+\frac{1}{3.5}+...+\frac{1}{95.97}\)
\(\Rightarrow A=\frac{1}{2}.\left(\frac{2}{1.3}+\frac{2}{3.5}+...+\frac{2}{95.97}\right)\)
\(\Rightarrow A=\frac{1}{2}.\left(1-\frac{1}{3}+\frac{1}{3}-\frac{1}{5}+...+\frac{1}{95}-\frac{1}{97}\right)\)
\(\Rightarrow A=\frac{1}{2}.\left(1-\frac{1}{97}\right)\)
\(\Rightarrow A=\frac{1}{2}.\frac{96}{97}\)
\(\Rightarrow A=\frac{48}{97}.\)
+ Thay A vào \(\left(1\right)\) ta được:
\(\frac{1}{99.97}-\frac{48}{97}\)
\(=\frac{1}{99.97}-\frac{48.99}{99.97}\)
\(=\frac{1-48.99}{99.97}\)
\(=-\frac{4751}{9603}.\)
Vậy \(\frac{1}{99.97}-\frac{1}{97.95}-\frac{1}{95.93}-...-\frac{1}{5.3}-\frac{1}{3.1}=-\frac{4751}{9603}.\)
Chúc bạn học tốt!
Đặt \(B=\frac{1}{99.97}-\frac{1}{97.95}-\frac{1}{95.93}-...-\frac{1}{5.3}-\frac{1}{3.1}\)
\(\Rightarrow B=\frac{1}{99.97}-\left(\frac{1}{1.3}+\frac{1}{3.5}+...+\frac{1}{95.97}\right)\)
Đặt \(A=\frac{1}{1.3}+\frac{1}{3.5}+...+\frac{1}{95.97}\)
\(\Rightarrow A=\frac{1}{2}.\left(\frac{2}{1.3}+\frac{2}{3.5}+...+\frac{2}{95.97}\right)\)
\(\Rightarrow A=\frac{1}{2}.\left(1-\frac{1}{3}+\frac{1}{3}-\frac{1}{5}+...+\frac{1}{95}-\frac{1}{97}\right)\)
\(\Rightarrow A=\frac{1}{2}.\left(1-\frac{1}{97}\right)\)
\(\Rightarrow A=\frac{1}{2}.\frac{96}{97}\)
\(\Rightarrow A=\frac{48}{97}\)
\(\Rightarrow B=\frac{1}{99.97}-\frac{48.99}{97.99}\)
\(\Rightarrow B=\frac{-4751}{9603}\)
