Question

[Lương Thu Trang]

GIÚP MÌNH ZỚI: 

Cho S=\(\frac{5}{2.3.4}\)+\(\frac{5}{3.4.5}\)+....+\(\frac{5}{98.99.100}\)+\(\frac{5}{99.100.101}\)

Answer 1

[Minh Triều]

làm tiếp theo

\(S=\frac{5}{2}.\left(\frac{2}{2.3.4}+\frac{2}{3.4.5}+...+\frac{2}{98.99.100}+\frac{2}{99.100.101}\right)\)

\(=\frac{5}{2}.\left(\frac{1}{2.3}-\frac{1}{3.4}+\frac{1}{3.4}-\frac{1}{4.5}+....+\frac{1}{98.99}-\frac{1}{99.100}+\frac{1}{99.100}-\frac{1}{100.101}\right)\)

\(=\frac{5}{2}.\left(\frac{1}{2.3}-\frac{1}{100.101}\right)\)

còn lại tự làm

Answer 2

[Phong Trần Nam]

\(S=\frac{5}{2\cdot3\cdot4}+\frac{5}{3\cdot4\cdot5}+......+\frac{5}{99\cdot100\cdot101}\)

\(S\frac{2}{5}=\frac{2}{2\cdot3\cdot4}+\frac{2}{3\cdot4\cdot5}+.....+\frac{2}{99\cdot100\cdot101}\)

\(\frac{2}{2\cdot3\cdot4}=\frac{1}{2\cdot3}-\frac{1}{3\cdot4}\)

\(\frac{2}{3\cdot4\cdot5}=\frac{1}{3\cdot4}-\frac{1}{4\cdot5}\)

.............

\(\frac{2}{99\cdot100\cdot101}=\frac{1}{99\cdot100}-\frac{1}{100\cdot101}\)

\(\Rightarrow S\frac{2}{5}=\frac{1}{2\cdot3}-\frac{1}{3\cdot4}+\frac{1}{3\cdot4}-\frac{1}{4\cdot5}+.........+\frac{1}{99\cdot100}-\frac{1}{100\cdot101}\)

\(\Rightarrow S\frac{2}{5}=\frac{1}{2\cdot3}-\frac{1}{100\cdot101}\)

\(\Rightarrow S\frac{2}{5}=\frac{1}{6}-\frac{1}{10100}\)

\(\Rightarrow S\frac{2}{5}=\frac{5047}{30300}\)

\(\Rightarrow S=\frac{5047}{30300}:\frac{2}{5}\)

\(\Rightarrow S=\frac{5047}{30300}\cdot\frac{5}{2}\)

\(\Rightarrow S=\frac{5047}{12120}\)