giải:

Đặt A=1.2.3+2.3.4+3.4.5+4.5.6+...+98.99.100

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

4A=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)

4A=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

4A=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

4A=98.99.100.101

=>A=98.99.100.101/4

\(\frac{1}{1.2.3}+\frac{1}{2.3.4}+\frac{1}{3.4.5}+...+\frac{1}{98.99.100}\)

\(=\frac{1}{2}.\left(\frac{2}{1.2.3}+\frac{2}{2.3.4}+\frac{2}{3.4.5}+...+\frac{2}{98.99.100}\right)\)

\(=\frac{1}{2}.\left(\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+\frac{1}{3.4}-\frac{1}{4.5}+...+\frac{1}{98.99}-\frac{1}{99.100}\right)\)

\(=\frac{1}{2}.\left(\frac{1}{1.2}-\frac{1}{99.100}\right)\)

\(=\frac{1}{2}.\left(\frac{1}{2}-\frac{1}{9900}\right)\)

\(=\frac{1}{2}.\frac{4949}{9900}\)

\(=\frac{4949}{19800}\)

Ai hỏi

\(2A=\frac{2}{1.2.3}+\frac{2}{2.3.4}+\frac{2}{3.4.5}+...+\frac{2}{98.99.100}\)

\(2A=\frac{3-1}{1.2.3}+\frac{4-2}{2.3.4}+\frac{5-3}{3.4.5}+...+\frac{100-98}{98.99.100}\)

\(2A=\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+\frac{1}{3.4}-\frac{1}{4.5}+...+\frac{1}{98.99}-\frac{1}{99.100}\)

\(2A=\frac{1}{2}-\frac{1}{99.100}=\frac{49}{99.100}\Rightarrow A=\frac{49}{2.99.100}\)

Đặt \(A=\frac{1}{1.2.3}+\frac{1}{2.3.4}+\frac{1}{3.4.5}+...+\frac{1}{98.99.100}\)

\(A=\frac{1}{2}\left(\frac{2}{1.2.3}+\frac{2}{2.3.4}+\frac{2}{3.4.5}+...+\frac{2}{98.99.100}\right)\)

\(A=\frac{1}{2}\left(\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+\frac{1}{3.4}-\frac{1}{4.5}+...+\frac{1}{98.99}-\frac{1}{99.100}\right)\)

\(A=\frac{1}{2}\left(\frac{1}{1.2}-\frac{1}{99.100}\right)\)

chỗ nãy rồi bạn tự tính tiếp

KQ la \(\frac{4949}{19800}\)ak cac ban

C = 1.2.3 + 2.3.3 + 3.3.4 + .... + 3.99.100 
Đặt M = 1.2.3 + 2.3.4 + 3.4.5 + .... + 99.100.101 
=> M - 3A = 1.2.3 - 1.2.3 + 2.3.(4-3) + 3.4 ( 5-3) + .... + 99.100 ( 101 -3) 
= 1.2.3 + 2.3.4 + .... + 98.99.100 
=> M -3A = M - 99.100.101 
=> A = 99.100.101/3 = 333300

D = 

Đặt A=1.3.5 + 3.5.7 + 5.7.9 + ................ + 95.97.99
 8A= 1.3.5.8 + 3.5.7.8 + 5.7.9.8 + ................ + 95.97.99.8
8A=1.3.5(7+1)+3.5.7(9-1)+5.7.9.(11-3)+.......+95.97.99(101-93)
8A=3.5.7+15+3.5.7.9-3.5.7+5.7.9.11-3.5.7.9+.......+95.97.99.101-93.95.97.99
8A=15+95.97.99.101
 A= \(\frac{15+95.97.99.101}{8}\)
 A=11517600

\(A=2.\left(\frac{1}{1.2.3}+\frac{1}{2.3.4}+\frac{1}{3.4.5}+...+\frac{1}{98.99.100}\right)\)

\(A=2.\left(\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+...+\frac{1}{98.99}-\frac{1}{99.100}\right)\)

\(A=2.\left(\frac{1}{1.2}-\frac{1}{99.100}\right)\)

\(A=2\cdot\frac{4949}{9900}=\frac{4949}{4950}\)

\(A=\frac{11}{1.2.3}+\frac{11}{2.3.4}+\frac{11}{3.4.5}+...+\frac{11}{98.99.100}\)

\(A=\frac{11}{2}.\left(\frac{2}{1.2.3}+\frac{2}{2.3.4}+\frac{2}{3.4.5}+...+\frac{2}{98.99.100}\right)\)

\(A=\frac{11}{2}.\left(\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+\frac{1}{3.4}-\frac{1}{4.5}+...+\frac{1}{98.99}-\frac{1}{99.100}\right)\)

\(A=\frac{11}{2}.\left(\frac{1}{1.2}-\frac{1}{99.100}\right)=\frac{11}{2}.\left(\frac{1}{2}-\frac{1}{9900}\right)=\frac{11}{2}.\left(\frac{4950}{9900}-\frac{1}{9900}\right)=\frac{11}{2}.\frac{4949}{9900}=\frac{4949}{1800}\)

A=1/1.2.3+1/2.3.4+...+1/98.99.100

2A=2/1.2.3+2/2.3.4+...+2/98.99.100

2A=1/1.2-1/2.3+1/2.3-1/3.4+...+1/98.99-1/99.100

2A=1/1.2-1/99.100

2A=1/2-1/9900

2A=4949/9900

A=4949/19800

A=1/1.2.3+1/2.3.4+...+1/98.99.100

2A=2/1.2.3+2/2.3.4+...+2/98.99.100

2A=1/1.2-1/2.3+1/2.3-1/3.4+...+1/98.99-1/99.100

2A=1/1.2-1/99.100

2A=1/2-1/9900

2A=4949/9900

A=4949/19800