Ta có :

\(S=\dfrac{2}{1.2.3}+\dfrac{2}{2.3.4}+\dfrac{2}{3.4.5}+...............+\dfrac{2}{2009.2010.2011}\)

\(S=\dfrac{1}{1.2}-\dfrac{1}{2.3}+\dfrac{1}{2.3}-\dfrac{1}{3.4}+.........+\dfrac{1}{2009.2010}-\dfrac{1}{2010.2011}\)

\(S=\dfrac{1}{1.2}-\dfrac{1}{2010.2011}\)

\(S=\dfrac{1}{2}-\dfrac{1}{4042110}\) \(< \dfrac{1}{2}\)

\(\Rightarrow S< Q\)

s=1/1*2-1/2*3+1/2*3-1/3*4+....+1/2009*2010-1/210*2011

 =1/1*2-1/2010*2011

<1/1*2

\(S=\frac{2}{1\cdot2\cdot3}+\frac{2}{2\cdot3\cdot4}+\frac{2}{3\cdot4\cdot5}+...+\frac{2}{2009\cdot2010\cdot2011}\)

\(S=\frac{1}{1\cdot2}-\frac{1}{2\cdot3}+\frac{1}{2\cdot3}-\frac{1}{3\cdot4}+\frac{1}{3\cdot4}-\frac{1}{4\cdot5}+...+\frac{1}{2009\cdot2010}-\frac{1}{2010\cdot2011}\)

\(S=\frac{1}{1\cdot2}-\frac{1}{2010\cdot2011}\)

\(S=\frac{1}{2}-\frac{1}{2010\cdot2011}< \frac{1}{2}\)

=> S < P

\(S=\dfrac{2}{1.2.3}+\dfrac{2}{2.3.4}+\dfrac{2}{3.4.5}+...+\dfrac{2}{2009.2010.2011}\)

\(=\dfrac{1}{1.2}-\dfrac{1}{2.3}+\dfrac{1}{2.3}-\dfrac{1}{3.4}+\dfrac{1}{3.4}-\dfrac{1}{4.5}+...+\dfrac{1}{2009.2010}-\dfrac{1}{2010.2011}\)

\(=\dfrac{1}{1.2}-\dfrac{1}{2010.2011}\)

\(=\dfrac{1}{2}-\dfrac{1}{4042110}< \dfrac{1}{2}\)

\(\Rightarrow\) \(S< P\)

Vậy \(S< P\)

1/2-1/2010.2011

cậu có biết tách ko?

nếu cậu biết tách ra thành cách hiệu thì sẽ làm được nhanh thôi

Câu 1: 

Đặt S = 1.2+2.3+3.4+...+30.31

  3 S  = 1.2.3+2.3.3+3.4.3+...+30.31.3

  3 S  = 1.2.(3-0) + 2.3.(4-1) + 3.4.(5-2) + ...+ 30.31.(32-29)

  3S  = 1.2.3 + 2.3.4-2.3 + 3.4.5-2.3.4 + ...+ 30.31.32-29.30.31

  3S= 30.31.32 

S= 30.31.32/3

Ta có: A = 1.2.3+3.4.5+5.6.7+...+99.100.101

A = 1.3 (5-3) + 3.5 (7-3) + 5.7 (9-3) + ............ + 99.101 (103 - 3)

A = (1.3.5 + 3.5.7 + 5.7.9 + .......... + 99.101.103) - (1.3.3 + 3.5.3 + ....... + 99.101.3)

A = (15+99.101.103.105) : 8 - 3.(1.3 + 3.5 +5.7 + ...... + 99.101)

A = 13517400 - 3.171650

A = 13002450

1.2.3.4+2.3.4.5+3.4.5.6+...+97.98.99.100

4S=(1.2.3+2.3.4+3.4.5+4.5.6+...+98.99.100). 4

4S=1.2.3(4-0)+2.3.4(5-1)+3.4.5(6-2)+4.5.6(7-3)+...+98.99.100(101-97)

4S=1.2.3.4+2.3.4.5-1.2.3.4+3.4.5.6-2.3.4.5+4.5.6.7-3.4.5.6+...98.99.100.101-97.98.99.100

4S=1.2.3.4-1.2.3.4+2.3.4.5-2.3.4.5+3.4.5.6-3.4.5.6+...+97.98.99.100-97.98.99.100+98+99.100+101

4S=98.99.100.101

Vậy S = 98.99.100.101/4 = 24497550