Question

Tính nhanh giá trị biểu thức sau :

\(\frac{4}{5.7}\)+ \(\frac{4}{7.9}\)+ ...+ \(\frac{4}{59.61}\)

Answer 1

Ta có:\(\frac{4}{5.7}+\frac{4}{7.9}+.....+\frac{4}{59.61}\)

\(\Rightarrow2.\left(\frac{2}{5.7}+\frac{2}{7.9}+......+\frac{2}{59.61}\right)\)

\(\Rightarrow2.\left(\frac{1}{5}-\frac{1}{7}+\frac{1}{7}-\frac{1}{9}+....+\frac{1}{59}-\frac{1}{61}\right)\)

\(\Rightarrow2.\left(\frac{1}{5}-\frac{1}{61}\right)\)

\(\Rightarrow\frac{112}{305}\)

Answer 2

\(\frac{4}{5.7}+\frac{4}{7.9}+...+\frac{4}{59.61}\)

\(=\frac{4.2}{5.7.2}+\frac{4.2}{7.9.2}+...+\frac{4.2}{59.61.2}\)

\(=\frac{4}{2}.\left(\frac{2}{5.7}+\frac{2}{7.9}+...+\frac{2}{59.61}\right)\)

\(=\frac{4}{2}\left(\frac{1}{5}-\frac{1}{7}+\frac{1}{7}-\frac{1}{9}+...+\frac{1}{59}-\frac{1}{61}_{ }\right)\)

\(=\frac{4}{2}.\left(\frac{1}{5}-\frac{1}{60}\right)\)

\(=\frac{4}{2}.\frac{11}{60}\)

\(=\frac{11}{30}\)

Answer 3

\(\frac{\text{4}}{5.7}\)+ \(\frac{4}{7.9}\)+...+ \(\frac{4}{59.60}\)

=\(\frac{4}{2}\).( \(\frac{2}{5.7}\)+\(\frac{2}{7.9}\)+...+\(\frac{2}{59.60}\))

=\(\frac{4}{2}\).(\(\frac{1}{5}\)-\(\frac{1}{7}\)+\(\frac{1}{7}\)-\(\frac{1}{9}\)+...+\(\frac{1}{59}\)-\(\frac{1}{60}\))

=\(\frac{4}{2}\).(\(\frac{1}{5}\)-\(\frac{1}{60}\))

=\(\frac{2}{5}\)-\(\frac{1}{30}\)

=\(\frac{12}{30}\)-\(\frac{1}{30}\)

=\(\frac{11}{30}\)

Answer 4

Đặt \(A=\frac{4}{5.7}+\frac{4}{7.9}+...+\frac{4}{59.61}\)

\(2A=4\left(\frac{1}{5.7}+\frac{1}{7.9}+...+\frac{1}{59.61}\right)\)

\(2A=4\left(\frac{1}{5}+\frac{1}{7}+\frac{1}{7}-\frac{1}{9}+...+\frac{1}{59}-\frac{1}{61}\right)\)

\(2A=4\left(\frac{1}{5}-\frac{1}{61}\right)\)

\(2A=4.\frac{56}{305}\)

\(2A=\frac{224}{305}\)

\(A=\frac{224}{305}:2=\frac{112}{305}\)

Answer 5

Đặt \(A=\frac{4}{5.7}+\frac{4}{7.9}+...+\frac{4}{59.61}\)

\(\Rightarrow A=2.\left(\frac{2}{5.7}+\frac{2}{7.9}+...+\frac{2}{59.61}\right)\)

\(\Rightarrow A=2.\left(\frac{1}{5}-\frac{1}{7}+\frac{1}{7}-\frac{1}{9}+...+\frac{1}{59}-\frac{1}{61}\right)\)

\(\Rightarrow A=2.\left(\frac{1}{5}-\frac{1}{61}\right)\)

\(\Rightarrow A=2.\frac{56}{305}\)

\(\Rightarrow A=\frac{112}{305}\)

Answer 6

\(=2.\left(\frac{2}{5.7}+\frac{2}{7.9}+...+\frac{2}{59.61}\right)\)

\(=2.\left(\frac{7-5}{5.7}+\frac{9-7}{7.9}+...+\frac{61-59}{59.61}\right)\)

\(=2.\left(\frac{1}{5}-\frac{1}{7}+\frac{1}{7}-\frac{1}{9}+...+\frac{1}{59}-\frac{1}{61}\right)\)

\(=2.\left(\frac{1}{5}-\frac{1}{61}\right)\)

\(=2.\left(\frac{61}{305}-\frac{5}{305}\right)\)

\(=2.\frac{56}{305}\)

\(=\frac{112}{305}\)