Đáp án:

 a)2003/2005

b)4949/9900

Giải thích các bước giải:

 a)M=1/3+1/6+1/10+1/15+....+1/2004.2005

M.1/2=1/6+1/12+1/20+1/30+....+1/2004.2005

M.1/2=1/2.3=1/3.4+1/4.5+1/5.6+....+1/2004.2005

M.1/2=1/2-1/3+1/3-1/4+1/4-1/5+1/5-1/6+....+1/2004-1/2005

M.1/2=1/2-1/2005

M.1/2=2003/4010

=>M=2003/2005

b)S=1/1.2.3+1/2.3.4+1/3.4.5+....+1/98.99.100

=2S=2x(1/1.2.3+1/2.3.4+1/3.4.5+...+1/98.99.100)

=2/1.2.3-2/2.3.4-2/3.4.5+....+2/98.99.100

=1/1.2-1/2.3+1/2.3-1/3.4+....+1/98.99-1/99.100

=1/1.2-1/99.100

=4950/9900-1/9900

=4949/9900

Giải thích các bước giải:

$M = \frac{1}{3} + \frac{1}{6} + \frac{1}{10} + ... + \frac{1}{2004.2005}$

⇔$\frac{1}{2}M = \frac{1}{6} + \frac{1}{12} + ... + \frac{1}{2004.2005}$

⇔$\frac{1}{2}M = \frac{1}{2.3} + \frac{1}{3.4} + ... + \frac{1}{2004.2005}$

⇔$\frac{1}{2}M = \frac{1}{2} - \frac{1}{3} + \frac{1}{3} - \frac{1}{4} +  ... + \frac{1}{2004} - \frac{1}{2005}$

⇔$\frac{1}{2}M = \frac{1}{2} - \frac{1}{2005}$

⇔$\frac{1}{2}M = \frac{2003}{4010}$

⇔$M = \frac{2003}{4010} : \frac{1}{2}$

⇔$M = \frac{2003}{2005}$