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2 tháng 8 2015

1.

\(A=\frac{1.2}{2.2}.\frac{2.3}{3.3}.\frac{3.4}{4.4}......\frac{2012.2013}{2013.2013}\)

\(A=\frac{1}{2}.\frac{2}{3}.\frac{3}{4}.........\frac{2012}{2013}\)

\(A=\frac{1.2.3.4.....2012}{2.3.4.5......2013}\)

\(A=\frac{1}{2013}\)

 

\(B=\frac{2012.2013-2012.2012}{2012.2011+2012.2}\)

\(B=\frac{2012\left(2013-2012\right)}{2012\left(2011+2\right)}\)

\(B=\frac{2013-2012}{2011+2}\)

\(B=\frac{1}{2013}\)

\(Vì:\frac{ 1}{2013}=\frac{1}{2013}\)

\(\Rightarrow\frac{1.2}{2.2}.\frac{2.3}{3.3}.\frac{3.4}{4.4}......\frac{2012.2013}{2013.2013}=\frac{2012.2013-2012.2012}{2012.2011+2012.2}\)

\(Hay: A=B\)

10 tháng 6 2018

\(A=\frac{1\times2}{2\times2}\times\frac{2\times3}{3\times3}\times\frac{3\times4}{4\times4}\times\frac{4\times5}{5\times5}\times...\times\frac{2012\times2013}{2013\times2013}\)

\(\Rightarrow A=\frac{1}{2}\times\frac{2}{3}\times\frac{3}{4}\times\frac{4}{5}\times...\times\frac{2012}{2013}\)

\(\Rightarrow A=\frac{1\times2\times3\times4\times...\times2012}{2\times3\times4\times5\times...\times2013}\)

\(\Rightarrow A=\frac{1}{2013}\)

\(B=\frac{2012\times2013-2012\times2012}{2012\times2011+2012\times2}\)

\(\Rightarrow B=\frac{2012\times\left(2013-2012\right)}{2012\times\left(2011+2\right)}\)

\(\Rightarrow B=\frac{2012\times1}{2012\times2013}\)

\(\Rightarrow B=\frac{1}{2013}\)

2 tháng 8 2016

A=1.2.3+2.3(4-1)+3.4(5-2)+4.5(6-3)+....+98.99(100-97)   "." la dau nhan

A=1.2.3+2.3.4-1.2.3+3.4.5-2.3.4+4.5.6-3.4.5+....+98.99.100-97.98.99

A=1.2.3+98.99.100

A= 970206

17 tháng 9 2017

Ta có : B = 1.2 + 2.3 + 3.4 + ..... + 98.99

=> 3B = 0.1.2 + 1.2.3 - 1.2.3 + ...... + 98.99.100

=> 3B = 98.99.100

=> B = \(\frac{98.99.100}{3}\)  = 323400

11 tháng 3 2016

\(=\frac{1}{2}-\frac{1}{100}=\frac{49}{100}\)

11 tháng 3 2016

\(\Leftrightarrow\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{99}-\frac{1}{100}\)

\(\Rightarrow\frac{1}{2}-0+0+...+0-\frac{1}{100}\)

\(\Rightarrow\frac{50}{100}-\frac{1}{100}=\frac{49}{100}\)

6 tháng 11 2017

\(\frac{1}{1x2}+\frac{1}{2x3}+\frac{1}{3x4}+\frac{1}{4x5}+...+\frac{1}{8x9}\)

=\(1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{8}-\frac{1}{9}\)

=\(1-\frac{1}{9}\)

=\(\frac{8}{9}\)

OK XONG NHỚ CHO MIK NHA

6 tháng 11 2017

\(\frac{1}{1\times2}+\frac{1}{2x3}+\frac{1}{3x4}+\frac{1}{4x5}+.......+\frac{1}{7x8}+\)\(\frac{1}{8x9}\)

=1-\(\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+.....+\frac{1}{8}-\frac{1}{9}\)

=1-\(\frac{1}{9}\)

=\(\frac{8}{9}\)

14 tháng 8 2016

\(\frac{1}{1x2}x\frac{4}{2x3}x\frac{9}{3x4}x...x\frac{10000}{100x101}=\frac{1x1}{1x2}x\frac{2x2}{2x3}x\frac{3x3}{3x4}x...x\frac{100x100}{100x101}\)

=\(\frac{1x2x3x...x100}{1x2x3x...x100}x\frac{1x2x3x...x100}{2x3x4x...x101}=1x\frac{1}{101}=\frac{1}{101}\)

18 tháng 10 2018

a, \(A=\frac{1}{2\cdot2}+\frac{1}{3\cdot3}+\frac{1}{4\cdot4}+...+\frac{1}{2011\cdot2011}\)

có :

\(\frac{1}{2\cdot2}< \frac{1}{1\cdot2}\)

\(\frac{1}{3\cdot3}< \frac{1}{2\cdot3}\)

\(\frac{1}{4\cdot4}< \frac{1}{3\cdot4}\)

...

\(\frac{1}{2011\cdot2011}< \frac{1}{2010\cdot2011}\)

nên :

\(A< \frac{1}{1\cdot2}+\frac{1}{2\cdot3}+\frac{1}{3\cdot4}+...+\frac{1}{2010\cdot2011}\)

\(\Rightarrow A< 1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{2010}-\frac{1}{2011}\)

\(\Rightarrow A< 1-\frac{1}{2011}\)

\(\Rightarrow A< \frac{2010}{2011}< 1\)

b, \(A=\frac{2010}{2011}=1-\frac{1}{2011}\) 

\(\frac{3}{4}=1-\frac{1}{4}\)

\(\frac{1}{4}>\frac{1}{2011}\)

nên :

\(A>\frac{3}{4}\)

19 tháng 3 2020

a, A bé hơn 1

b, A bé hơn 3/4

7 tháng 8 2016

\(\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{x\left(x+1\right)}=\frac{499}{500}\)

\(1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{x}-\frac{1}{x+1}=\frac{499}{500}\)

\(1-\frac{1}{x+1}=\frac{499}{500}\)

\(\frac{1}{x+1}=1-\frac{499}{500}=\frac{1}{500}\)

=> x + 1 = 500

=> x = 500 - 1

=> x = 499

Vậy x = 499

7 tháng 8 2016

1/1.2 + 1/2.3 + 1/3.4 +...+ 1/x.(x+1)=499/500

1 - 1/2 + 1/2 -1/3 + 1/3 - 1/4 +...+ 1/x -1/(x+1) =499/500

1-1/(x+1)=499/500

=>x/(x+1)=499/500

=>x=499

9 tháng 9 2017

\(A=\frac{5}{1.2}+\frac{5}{2.3}+...+\frac{5}{7.8}\)

\(\Rightarrow5A=\frac{1}{1.2}+\frac{1}{2.3}+...+\frac{1}{7.8}\)

\(\Rightarrow5A=1.\left(1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-...-\frac{1}{8}\right)\)

\(\Rightarrow5A=1-\frac{1}{8}\)

\(\Rightarrow A=\left(1-\frac{1}{8}\right).\frac{1}{5}=\frac{7}{40}\)

9 tháng 9 2017

\(A=\frac{5}{1.2}+\frac{5}{2.3}+...+\frac{5}{7.8}\)

\(A=5\left(\frac{1}{1.2}+\frac{1}{2.3}+...+\frac{5}{7.8}\right)\)

\(A=5\left(1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...+\frac{1}{7}-\frac{1}{8}\right)\)

\(A=5\left(1-\frac{1}{8}\right)\)

\(A=5.\frac{7}{8}\)

\(A=\frac{38}{8}\)

15 tháng 6 2016

1/1×2 + 1/2×3 + 1/3×4 + 1/4×5 + ... + 1/99×100

= 1 - 1/2 + 1/2 - 1/3 + 1/3 - 1/4 + 1/4 - 1/5 + ... + 1/99 - 1/100

= 1 - 1/100

= 99/100