\(S=\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{2011.2012}\)

\(S=\frac{1}{1}-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{2011}-\frac{1}{2012}\)

\(S=1-\frac{1}{2012}\)

\(S=\frac{2011}{2012}\)

Chúc bạn học tốt nha !!!

=1-1/2+1/2-1/3+1/3-1/4+...+1/2011-1/2012

= 1-1/2012

= 2011/2012

S=9/10

bằng 9/10 đó bạn

\(S=\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+.......+\frac{1}{2015.2016}\)

\(S=\frac{1}{1}-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+.......+\frac{1}{2015}-\frac{1}{2016}\)

\(S=\frac{1}{1}-\frac{1}{2016}=\frac{2015}{2016}\)

làm r sao cứ đăng hoài vậy?

=> 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

A = 1/1 - 1/2 + 1/2 - 1/3 + 1/3 - 1/4 +...+ 1/2011 - 1/2012

A = 1 - 1/2012

A = 2011/2012

B = 1/2 - 1/4 + 1/4 - 1/6 + 1/6 - 1/8 +...+ 1/2010 - 1/2012

B = 1/2 - 1/2012

B = 1005/2012

a) \(A=\dfrac{1}{1\cdot2}+\dfrac{1}{2\cdot3}+\dfrac{1}{3\cdot4}+...+\dfrac{1}{2011\cdot2012}\)

\(A=1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+...+\dfrac{1}{2011}-\dfrac{1}{2012}\)

\(A=1-\dfrac{1}{2012}\)

\(A=\dfrac{2011}{2012}\)

b) \(B=\dfrac{1}{2\cdot4}+\dfrac{1}{4\cdot6}+\dfrac{1}{6\cdot8}+...+\dfrac{1}{2010\cdot2012}\)

\(B=\dfrac{1}{2}\cdot\left(\dfrac{2}{2\cdot4}+\dfrac{2}{4\cdot6}+\dfrac{2}{6\cdot8}+...+\dfrac{2}{2010\cdot2012}\right)\)

\(B=\dfrac{1}{2}\cdot\left(\dfrac{1}{2}-\dfrac{1}{4}+\dfrac{1}{4}-\dfrac{1}{6}+\dfrac{1}{6}-\dfrac{1}{8}+...+\dfrac{1}{2010}-\dfrac{1}{2012}\right)\)

\(B=\dfrac{1}{2}\cdot\left(\dfrac{1}{2}-\dfrac{1}{2012}\right)\)

\(B=\dfrac{1}{2}\cdot\dfrac{1005}{2012}\)

\(B=\dfrac{1005}{4024}\)

=> 3S = 1.2.3 + 2.3.3 + 3.4.3 + ... + 99.100.3

= 1.2.(3 - 0) + 2.3.(4 - 1) + 3.4.(5 - 2) + ... + 99.100.(101 - 98)

= 1.2.3 - 0 + 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

ta có \(3S=1\cdot2\cdot3+2\cdot3\cdot3+.....+99\cdot100\cdot3\)

\(3S=1\cdot2\cdot3+2\cdot3\cdot\left(4-1\right)....+99\cdot100\cdot\left(101-98\right)\)

\(3S=1\cdot2\cdot3-1\cdot2\cdot3+2\cdot3\cdot4-......-98\cdot99\cdot100+99\cdot100\cdot101\)

\(3S=99.100.101\)

\(S=\frac{99\cdot100\cdot101}{3}\)

S=...

3S=1.2.3+2.3.3+3.4.3+4.5.3+...+99.100.3

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

3S=99.100.101

S=33.100.101

S=333300

Vậy S=333300