\(S=\left(\frac{1}{1.3}+\frac{1}{3.5}+\frac{1}{5.7}+\frac{1}{7.9}\right)-\left(\frac{1}{2.4}+\frac{1}{4.6}+\frac{1}{6.8}+\frac{1}{8.10}\right)\)

=>\(S=\frac{1}{2}.\left(\frac{2}{1.3}+\frac{2}{3.5}+\frac{2}{5.7}+\frac{2}{7.9}\right)-\frac{1}{2}.\left(\frac{2}{2.4}+\frac{2}{4.6}+\frac{2}{6.8}+\frac{2}{8.10}\right)\)

=>\(S=\frac{1}{2}.\left(1-\frac{1}{3}+\frac{1}{3}-\frac{1}{5}+\frac{1}{5}-\frac{1}{7}+\frac{1}{7}-\frac{1}{9}\right)-\frac{1}{2}.\left(\frac{1}{2}-\frac{1}{4}+\frac{1}{4}-\frac{1}{6}+\frac{1}{6}-\frac{1}{8}+\frac{1}{8}-\frac{1}{10}\right)\)

=>\(S=\frac{1}{2}.\left(1-\frac{1}{9}\right)-\frac{1}{2}.\left(\frac{1}{2}-\frac{1}{10}\right)\)

=>\(S=\frac{1}{2}.\frac{8}{9}-\frac{1}{2}.\frac{2}{5}\)

=>\(S=\frac{4}{9}-\frac{1}{5}\)

=>\(S=\frac{11}{45}\)

lê chí cường dung 

= 1/1 - 1/5 + 1/5 -1/6 + 1/6 - 1/7 + 1/7 - 1/8 + 1/8-1/9 +...+ 1/2004 - 1/2005

= 1/1 - 1/2005

= 2004/2005

ai giúp mình vời

thế kết quả thứ 2 của mình là 2006/2005

\(\frac{6^2}{5\cdot7}\cdot\frac{7^2}{6\cdot8}\cdot\frac{8^2}{7\cdot9}\cdot\frac{9^2}{8\cdot10}\)

=\(\frac{6\cdot6}{5\cdot7}\cdot\frac{7\cdot7}{6\cdot8}\cdot\frac{8\cdot8}{7\cdot9}\cdot\frac{9\cdot9}{8\cdot10}\)

=\(\frac{6\cdot7\cdot8\cdot9}{5\cdot6\cdot7\cdot8}\cdot\frac{6\cdot7\cdot8\cdot9}{7\cdot8\cdot9\cdot10}\)

=\(\frac{9}{5}\cdot\frac{3}{5}\)=\(\frac{27}{25}\)

**** MIK

\(=\frac{6.6}{5.7}.\frac{7.7}{6.8}.\frac{8.8}{7.9}.\frac{9.9}{8.10}\)

\(=\frac{6.6.7.7.8.8.9.9}{5.6.7.8.9.10.8.7}\)

\(=\frac{27}{25}\)

Bài làm

\(A=\frac{2011.2012-1}{2011.2012}\) và \(B=\frac{2012.2013-1}{2012.2013}\)

Ta có:

\(A=\frac{2011.2012-1}{2011.2012}\)

\(A=\frac{2011.2012-1.1-1.1}{2011.2012}\)

\(A=\frac{2011.2012-1.\left(1-1\right)}{2011.2012}\)

\(A=\frac{2011.2012-1.0}{2011.2012}\)

\(A=\frac{2011.2012-0}{2011.2012}\)

\(A=\frac{2011.2012}{2011.2012}\)

\(A=1\)

\(B=\frac{2012.2013-1}{2012.2013}\)

\(B=\frac{2012.2013-1.1-1.1}{2012.2013}\)

\(B=\frac{2012.2013-1.\left(1-1\right)}{2012.2013}\)

\(B=\frac{2012.2013-1.0}{2012.2013}\)

\(B=\frac{2012.2013-0}{2012.2013}\)

\(B=\frac{2012.2013}{2012.2013}\)

\(B=1\)

Vì 1 = 1

=> A = B

Hay

\(A=\frac{2011.2012-1}{2011.2012}\)=  \(B=\frac{2012.2013-1}{2012.2013}\)

Vậy \(A=\frac{2011.2012-1}{2011.2012}\)= \(B=\frac{2012.2013-1}{2012.2013}\)

# Chúc bạn học tốt #

Ta có : A =( 2011.2012-1)/(2011.2012) = (2011.2012)/(2011.2012) - 1/(2011.2012) = 1 - (1/2011.2012)

           B =( 2012.2013-1)/(2012.2013) = (2012.2013)/(2012.2013) - 1/(2012.2013) = 1 - (1/2012.2013)

Ta thấy : 1/(2011.2012)>1/(2012.2013)(vì chung tử số là 1 , mẫu số : 2011.2012 < 2012.2013)

Suy ra , 1-(1/2011.2012)<1-(1/2012.2013)

Suy tiếp :        A              <           B

Bài làm:

Ta có: \(S=\frac{1}{2.2}+\frac{1}{3.3}+\frac{1}{4.4}+...+\frac{1}{9.9}\)

\(>\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+..+\frac{1}{9.10}\)

\(=\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+...+\frac{1}{9}-\frac{1}{10}\)

\(=\frac{1}{2}-\frac{1}{10}=\frac{2}{5}\)\(\Rightarrow\frac{2}{5}< S\)

Cái còn lại tự CM

\(MS=2011.2013+2012\)

\(=\left(2012-1\right).2013+2012\)

\(=2012.2013-2013+2012\)

\(=2013.2012-1\)

\(=TS\)

Vậy phân số đã cho bằng 1.

Trả lời:

\(\frac{2013.2012-1}{2011.2013+2012}=\frac{2013.\left(2011+1\right)-1}{2011.2013+2012}\)

                                        \(=\frac{2011.2013+2013-1}{2011.2013+2012}\)

                                        \(=\frac{2011.2013+2012}{2011.2013+2012}\)

                                        \(=1\)

Học tốt

\(\frac{2013.2012-1}{2011.2013+2012}\)

\(=\frac{2013.2012-1}{2011.2013+2013-1}\)

\(=\frac{2013.2012-1}{2012.2013-1}\)

\(=1\)

a.\(\frac{3\cdot4\cdot7}{12\cdot8\cdot9}\)= \(\frac{3\cdot4\cdot7}{3\cdot4\cdot8\cdot9}\)= \(\frac{7}{72}\) 

b. \(\frac{4\cdot5\cdot6}{12\cdot10\cdot8}\)= \(\frac{4\cdot5\cdot2\cdot3}{3\cdot4\cdot5\cdot2\cdot8}\)= \(\frac{1}{8}\) 

c.\(\frac{5\cdot6\cdot7}{12\cdot14\cdot15}\)= \(\frac{5\cdot6\cdot7}{2\cdot6\cdot2\cdot7\cdot3\cdot5}\)= \(\frac{1}{12}\)

A= 1/2.2 + 1/3.3 + 1/4.4 + 1/5.5 + 1/6.6 + 1/7.7 + 1/8.8 + 1/9.9

Vì 1/2.2 > 1/2.3; 1/3.3 > 1/3.4 ; 1/5.5 > 1/5.6;...... nên 

1/2.2 +1/3.3 + 1/4.4 + 1/5.5 + 1/6.6 + 1/7.7 + 1/8.8 + 1/9.9 > 1/2.3 + 1/3.4 + 1/4.5 + 1/5.6 + 1/6.7 + 1/7.8 + 1/8.9 + 1/9.10

Ta có: 1/2.3 + 1/3.4 + 1/4.5 + 1/5.6 + 1/6.7 + 1/7.8 + 1/8.9 + 1/9.10

= 1/2-1/3 + 1/3 -1/4 + 1/4-1/5+...+1/9-1/10

= 1/2- 1/10

= 2/5

Vì A < 2/5 mà 2/5 <7/8 nên 2/5 < A < 7/8

Vậy....

13/30 nhé bạn

Kêt quả bằng 13/30

ta có:(3+4)/4=3/(3*4)+4/(3*4) =1/4+1/3 chứng minh tương tự,cộng vế với vế, ta được kết quả là 13/30

a, \(\frac{3.4.7}{12.8.9}\)= \(\frac{3.4.7}{3.4.8.9}\)= \(\frac{7}{72}\)

b, \(\frac{4.5.6}{12.10.8}\)= \(\frac{4.5.6}{3.4.2.5.8}\)= \(\frac{1}{8}\)

c, \(\frac{5.6.7}{12.14.15}\)= \(\frac{5.6.7}{2.6.2.7.3.5}\)= \(\frac{1}{12}\)