Question

Answer 1

1:

\(=\dfrac{1}{2}\cdot\left(\dfrac{2}{3\cdot5}+\dfrac{2}{5\cdot7}+...+\dfrac{2}{11\cdot13}\right)\)

=1/2*(1/3-1/5+1/5-1/7+...+1/11-1/13)

=1/2*10/39=5/39

2: \(=\dfrac{1}{3}\left(\dfrac{3}{1\cdot4}+\dfrac{3}{4\cdot7}+...+\dfrac{3}{13\cdot16}\right)\)

=1/3(1-1/4+1/4-1/7+...+1/13-1/16)

=15/16*1/3=5/16

3: \(=\dfrac{1}{4}\left(\dfrac{4}{1\cdot5}+\dfrac{4}{5\cdot9}+...+\dfrac{4}{17\cdot21}\right)\)

=1/4(1-1/5+1/5-1/9+...+1/17-1/21)

=20/21*1/4=5/21