Mình không chép đề bài nhé :
Gọi biểu thức là A :
Ta có : 2A=\(\frac{2}{1.2.3}+\frac{2}{2.3.4}+\frac{2}{3.4.5}+...+\frac{2}{48.49.50}\)
= \(\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+\frac{1}{3.4}-\frac{1}{4.5}+...+\frac{1}{48.49}-\frac{1}{49.50}\)
=\(\frac{1}{1.2}-\frac{1}{49.50}\)( Rút gọn đi ta được cái này )
=1/2 - 1/2450
Vậy A = (1/2 - 1/2450):2
 

\(\frac{1}{1.2.3}+\frac{1}{2.3.4}+...+\frac{1}{48.49.50}\)

= \(\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+...+\frac{1}{48.49}-\frac{1}{49.50}\)

= \(\frac{1}{1.2}-\frac{1}{49.50}\)

= \(\frac{1}{2}-\frac{1}{2450}\)

= \(\frac{612}{1225}\)

đặt

\(A=\frac{1}{1.2.3}+\frac{1}{2.3.4}+...+\frac{1}{48.49.50}\)

\(\Rightarrow2A=\frac{2}{1.2.3}+\frac{2}{2.3.4}+...+\frac{2}{48.49.50}\)

\(\Rightarrow\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+...+\frac{1}{48.49}-\frac{1}{49.50}\)

\(\Rightarrow\frac{1}{1.2}-\frac{1}{49.50}\)

\(\Rightarrow\frac{1}{2}-\frac{1}{2450}=\frac{621}{1225}\)

\(\Rightarrow A=\frac{306}{1225}\)

=>2A  = \(\frac{2}{1.2.3}+\frac{2}{2.3.4}+\frac{2}{3.4.5}+...+\frac{2}{48.49.50}\)

=\(\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+\frac{1}{3.4}+...+\frac{1}{48.49}-\frac{1}{49-50}\)

= \(\frac{1}{1.2}-\frac{1}{49.50}\)= \(\frac{1}{2}-\frac{1}{2450}=\frac{612}{1225}\)

\(=\frac{1}{2}\left(\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+...+\frac{1}{48.49}-\frac{1}{49.50}\right)\\ =\frac{1}{2}\left(\frac{1}{2}-\frac{1}{2450}\right)\)

\(=\frac{1}{2}.\frac{612}{1225}\\ =\frac{306}{1225}\)(mà đây là toán 6 mà :V)

\(A=\frac{1}{1.2.3}+\frac{1}{2.3.4}+\frac{1}{3.4.5}+...+\frac{1}{48.49.50}\)

\(A=\frac{1}{2}.\left(\frac{1}{1.2}-\frac{1}{2.3}\right)+\frac{1}{2}.\left(\frac{1}{2.3}-\frac{1}{3.4}\right)+...+\frac{1}{2}.\left(\frac{1}{48.49}-\frac{1}{49.50}\right)\)

\(A=\frac{1}{2}.\left(\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+...+\frac{1}{48.49}-\frac{1}{49.50}\right)\)

\(A=\frac{1}{2}.\left(\frac{1}{1.2}-\frac{1}{49.50}\right)\)

\(A=\frac{1}{2}.\left(\frac{1}{2}-\frac{1}{2450}\right)\)

\(A=\frac{1}{2}.\frac{612}{1225}=\frac{306}{1225}\)

\(A=\frac{1}{2}.\left(\frac{1}{1.2}-\frac{1}{1.3}\right)+\frac{1}{2}.\left(\frac{1}{2.3}\right)-\frac{1}{3.4}+...\frac{1}{2}\left(\frac{1}{48.49}\right)\)

\(A=\frac{1}{2}.\left(\frac{1}{2}-\frac{1}{6}+\frac{1}{6}-\frac{1}{12}+...\frac{1}{48.49}-\frac{1}{49.50}\right)\)

\(A=\left(\frac{1}{2}.\frac{1}{49.50}\right)\)

\(A=\frac{1}{2}.\left(\frac{1}{2}-\frac{1}{2450}\right)=\frac{1}{2}.\frac{1}{1225}=\frac{306}{1225}\)

\(B=-\frac{3}{5}\left(\frac{3}{8}-2+\frac{5}{8}\right)\)

\(B=-\frac{3}{5}.\left(-1\right)=\frac{3}{5}\)

\(C=\frac{8}{5}.\frac{3}{4}-\left(\frac{11}{20}-\frac{1}{4}\right).\frac{7}{3}\)

\(C=\frac{6}{5}-\frac{3}{10}.\frac{7}{3}\)

\(C=\frac{6}{5}-\frac{7}{10}=\frac{1}{2}\)

\(Z=\frac{1}{1.2.3}+\frac{1}{2.3.4}+\frac{1}{3.4.5}+...+\frac{1}{48.49.50}\)

\(=\frac{1}{2}\left(\frac{2}{1.2.3}+\frac{2}{2.3.4}+\frac{2}{3.4.5}+...+\frac{2}{48.49.50}\right)\)

\(=\frac{1}{2}\left(\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+\frac{1}{3.4}-\frac{1}{4.5}+...+\frac{1}{48.49}-\frac{1}{49.50}\right)\)

\(=\frac{1}{2}\left(\frac{1}{1.2}-\frac{1}{49.50}\right)=\frac{1}{2}\left(\frac{1}{2}-\frac{1}{2450}\right)=\frac{1}{2}\left(\frac{2450}{2450}-\frac{1}{2450}\right)\)

\(=\frac{1}{2}.\frac{2449}{2450}=\frac{2449}{4900}\)

Z = 1/1.2.3 + 1/2.3.4 + 1/3.4.5 + ... + 1/98.99.100 
Áp dụng phương pháp khử liên tiếp: viết mỗi số hạng thành hiệu của hai số sao cho số trừ ở nhóm trước bằng số bị trừ ở nhóm sau. 
Ta xét: 
1/1.2 - 1/2.3 = 2/1.2.3; 1/2.3 - 1/3.4 = 2/2.3.4;...; 1/98.99 - 1/99.100 = 2/98.99.100 
tổng quát: 1/n(n+1) - 1/(n+1)(n+2) = 2/n(n+1)(n+2). Do đó: 
2Z = 2/1.2.3 + 2/2.3.4 + 2/3.4.5 +...+ 2/98.99.100 
= (1/1.2 - 1/2.3) + (1/2.3 - 1/3.4) +...+ (1/98.99 - 1/99.100) 
= 1/1.2 - 1/2.3 + 1/2.3 - 1/3.4 + ... + 1/98.99 - 1/99.100 
= 1/1.2 - 1/99.100 
= 1/2 - 1/9900 
= 4950/9900 - 1/9900 
= 4949/9900. 
Vậy Z = \(\frac{4949}{9900}\)

A=1/2{(1/1*2-1/2*3)+(1/2*3-1/3*4)+(1/3*4-1/4*5)+...+(1/18*19-1/19*20)}
  =1/2{1/1*2-1/19*20}
  =1/2*189/380
  =189/760
vì 189/760<1/4
nên A=...<1/4