Ta có : \(\frac{1}{1.2}\)+  \(\frac{1}{2.3}\)+\(\frac{1}{3.4}\)+\(\frac{1}{4.5}\)+\(\frac{1}{5.6}\)+\(\frac{1}{6.7}\)

= 1  - \(\frac{1}{2}\)+\(\frac{1}{2}\)-\(\frac{1}{3}\)+\(\frac{1}{3}\)-\(\frac{1}{4}\)+\(\frac{1}{4}\)-\(\frac{1}{5}\)+\(\frac{1}{5}\)-\(\frac{1}{6}\)+\(\frac{1}{6}\)-\(\frac{1}{7}\)

= 1 - \(\frac{1}{7}\)=  \(\frac{6}{7}\)

=1-1/2+1/2-1/3+1/3-1/4+...+1/6-1/7=1-1/7=6/7

1/1 -1/2 +1/2 -1/3 +1/3 -1/4 +1/4 -1/5 +1/5 -1/6 +1/6 -1/7

1/1- 1/7 =6/7

3A =1.2.3 +2.3.(4-1) +3.4.(5-2) +4.5.(6-3)....+199.200.(201 -198)

    = 1.2.3+2.3.4 -1.2.3 +3.4.5- 2.3.4 + 4.5.6 - 3.4.5 +......+ 199.200.201 -198.199.200

 3A =199.200.201 

A=199.200.67 =254600

`S = 1.2 + 2.3 + 3.4 + 4.5 + ... + 99.100.`

`3S =  1.2.3 + 2.3.(4-1) + 3.4.(5-4) + 4.5.(6-3) + ... + 99.100.(101-98)`

`3S =  1.2.3 + 2.3.4-1.2.3 + 3.4.5-4.5.6 + 4.5.6-3.4.5 + ... + 99.100.101-98.99.100`

`3S =  99.100.101`

`S = 33.100.101`

`S = 333300`

3S=1.2(3-0)+2.3(4-1)+.....+99.100(101-98)

=1.2.3-0.1.2+2.3.4-1.2.3+4.5.6-2.3.4+....+99.100.101-98-99-100

=99.100.101

S=33.100.101

=333300

 S=1.2+ 2.3+4,5.......+99.100 
Nhân cả 2 vế với 3, ta được: 
3S=1.2.3+ 2.3.3+ 3.4.3+ 4.5.3+...... 99.100.3 
= 1.2.3 + 2.3(4-1) + 3.4.(5-2) +...+ 99.100.(101-98) 
= 1.2.3 + 2.3.4 -1.2.3 + 3.4.5-2.3.4 +...+ 99.100.101-98.99.100 
= 99.100.101 
----> S = (99.100.101):3 
 S= 333300 
Vậy A=333300 

S = 1.2 + 2.3 + 3.4 + 4.5 +...+ 99.100

S = 1.100

S = 100

A=1.2+2.3+...+49.50

3A=1.2.3+2.3.3+...+49.50.3

3A=1.2.(4-1)+2.3.(5-2)+....+49.50.(51-48)

3A=1.2.4-1.2.1+2.3.5-2.3.2+...+49.50.51-49.50.48

3A=49.50.51

=>A=49.25.51

=>A=62475

A=1.2+2.3+...+49.50

3A=1.2.3+2.3.3+...+49.50.3

3A=1.2.(4-1)+2.3.(5-2)+....+49.50.(51-48)

3A=1.2.4-1.2.1+2.3.5-2.3.2+...+49.50.51-49.50.48

3A=49.50.51

=>A=49.25.51

=>A=62475

3S=(1.2+2.3+3.4+...+49.50).3

3S=1.2.3+2.3.3+3.4.3+...+49.50.3

3S=1.2.3+2.3.(4-1)+3.4.(5-2)+...+49.50.(51-48)

3S=1.2.3+2.3.4-1.2.3+3.4.5-2.3.4+...+49.50.51-48.49.50

3S=49.50.51

S=17.49.50

=> 3S = 1.2.3 + 2.3.3 + 3.4.3 + .... + 2011.2012.3

=> 3S = 1.2.3 + 2.3.( 4 - 1 ) + 3.4.( 5 - 2 ) + .... + 2011.2012.( 2013 - 2010 )

=> 3S = 1.2.3 + 2.3.4 - 1.2.3 + .... + 2011.2012.2013 - 2010.2011.2012

=> 3S = ( 1.2.3 - 1.2.3 ) + ( 2.3.4 - 2.3.4 ) + .... + ( 2010.2011.2012 - 2010.2011.2012 ) + 2011.2012.2013

=> 3S = 2011.2012.2013

=> S = ( 2011.2012.2013 ) : 3

3S=1.2.3+2.3.(4-1)+...............+2011.2012.(2013-2010)

3S=1.2.3+2.3.4-1.2.3+...............+2011.2012.2013-2010.2011.2012

3S=2011.2012.2013

S=2011.2012.2013:3

S=2714954572

S=2714954572

S=1.2+2.3+3.4+...+99.100

3S=1.2.3+2.3.(4-1)+3.4.(5-2)+...+99.100.(101-98)

3S=1.2.3+2.3.4-1.2.3+3.4.5-2.3.4+...+99.100.101-98.99.100

3S=99.100.101

S=(99.100.101):3=333300

           S=1.2+2.3+3.4+4.5+...+98.99+99.100

suy ra :3S=1.2.3+2.3.3+3.4.3+4.5.3+...+98.99.3+99.100.3

            3S=1.2.(3-0)+2.3.(4-1)+3.4.(5-2)+4.5.(6-3)+...+98.99.(100-97)+99.100.(101-98)

           3S=1.2.3.0+2.3.4-1.2.3+3.4.5-2.3.4+4.5.6-3.4.5+...+98.99.100-97.98.99+99.100.101-98.99.100

           3S=99.100.101

Suy ra :S=99.100.10:3=333300

vậy S=333300

ko bit

=333300