Question

5/1.3 + 5/3.5 + ... + 5/99.101 = ?

Answer 1

\(\frac{5}{1.3}+\frac{5}{3.5}+...+\frac{5}{99.101}=\frac{5}{2}.\left(\frac{2}{1.3}+\frac{2}{3.5}+...+\frac{2}{99.101}\right)\)

\(=\frac{5}{2}.\left(1-\frac{1}{3}+\frac{1}{3}-\frac{1}{5}+...+\frac{1}{99}-\frac{1}{101}\right)=\frac{5}{2}.\left(1-\frac{1}{101}\right)=\frac{5}{2}.\frac{100}{101}=\frac{250}{101}\)

Answer 2

=\(\frac{5.2}{1.3.2}+\frac{5.2}{3.5.2}+...+\frac{5.2}{99.101.2}\)

=\(\frac{5}{2}.\left(\frac{2}{1.3}+\frac{2}{3.5}+...+\frac{2}{99.101}\right)\)

=\(\frac{5}{2}.\left(1-\frac{1}{3}+\frac{1}{3}-\frac{1}{5}+...+\frac{1}{99}-\frac{1}{101}\right)\)

=\(\frac{5}{2}.\left(1-\frac{1}{101}\right)\)

=\(\frac{5}{2}.\frac{100}{101}\)

=\(\frac{250}{101}\)