B = 2/5x7 + 3/7x10 + 4/10x14 + 7/14x21 + 9/21x30

B = 1/5 - 1/7 + 1/7 - 1/10 + 1/10 - 1/14 + 1/14 - 1/21 + 1/21 - 1/30

B = 1/5 - 1/30

B = 6/30 - 1/30

B = 5/30 = 1/6

B = 2/5x7 + 3/7x10 + 4/10x14 + 7/14x21 + 9/21x30

B = 1/5 - 1/7 + 1/7 - 1/10 + 1/10 - 1/14 + 1/14 - 1/21 + 1/21 - 1/30

B = 1/5 - 1/30

B = 6/30 - 1/30

B = 5/30 = 1/6

soyeon làm sai vì nó là 2/5x7 chứ có phải là 1/5x7 đâu mà lại tách ra như thế?

hình như có vấn đề ở 9/14 x 21 đáng lẽ là 9/14 x 23 chứ 

a: Ta có: \(A=\frac{1}{2\cdot3}+\frac{1}{3\cdot4}+\frac{1}{4\cdot5}+\frac{1}{5\cdot6}\)

\(=\frac12-\frac13+\frac13-\frac14+\frac14-\frac15+\frac15-\frac16\)

\(=\frac12-\frac16=\frac36-\frac16=\frac26=\frac13\)

b: Ta có: \(B=\frac{3}{7\cdot10}+\frac{24}{10\cdot14}-\frac{35}{14\cdot21}+\frac{49}{21\cdot70}\)

\(=\frac17-\frac{1}{10}+\frac{1}{10}+\frac{1}{14}-\frac{1}{14}-\frac{1}{21}+\frac{1}{21}-\frac{1}{70}\)

\(=\frac17-\frac{1}{70}=\frac{9}{70}\)

\(a,\frac{5}{1.3}+\frac{5}{3.5}+\frac{5}{5.7}+...+\frac{5}{43.45}=\frac{5}{2}\left(\frac{2}{1.3}+\frac{2}{3.5}+\frac{2}{5.7}+...+\frac{2}{43.45}\right)=\frac{5}{3}\left(1-\frac{1}{3}+\frac{1}{3}-\frac{1}{5}+\frac{1}{5}-\frac{1}{7}+...+\frac{1}{43}-\frac{1}{45}\right)=\frac{5}{3}.\frac{44}{45}=\frac{44}{27}\)

Bài 2 :

Ta có :

\(A=\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{32}=\left(1-\frac{1}{2}\right)+\left(\frac{1}{2}-\frac{1}{4}\right)+\left(\frac{1}{4}-\frac{1}{8}\right)+\left(\frac{1}{8}-\frac{1}{16}\right)+\left(\frac{1}{16}-\frac{1}{32}\right)\)

    \(=1-\frac{1}{32}\)

và \(B=\frac{2003}{2004}=1-\frac{1}{2004}\)

Vì \(\frac{1}{32}>\frac{1}{2004}\) nên A < B

A<B