\(\frac{1}{1.2.3}+\frac{1}{2.3.4}+...+\frac{1}{49.50.51}\)

\(=\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.5}+...+\frac{1}{49.50}-\frac{1}{50.51}\)

\(=\frac{1}{2}-\frac{1}{50.51}\)

\(=\frac{1}{2}-\frac{1}{2550}=\frac{637}{1275}\)

Gọi A là tổng dãy phân số trên

Ta có :

\(A=\frac{1}{1.2.3}+\frac{1}{2.3.4}+...+\frac{1}{49.50.51}\)

\(2A=\frac{2}{1.2.3}+\frac{2}{2.3.4}+...+\frac{2}{49.50.51}\)

Ta thấy:

\(\frac{2}{1.2.3}=\frac{1}{1.2}-\frac{1}{2.3};\frac{2}{2.3.4}=\frac{1}{2.3}-\frac{1}{3.4};...;\frac{2}{49.50.51}=\frac{2}{49.50}-\frac{2}{50.51}\text{​​}\)

\(\Rightarrow2A=\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+...+\frac{1}{49.50}-\frac{1}{50.51}\)

\(\Rightarrow2A=\frac{1}{1.2}-\frac{1}{50.51}\)

\(\Rightarrow2A=\frac{1}{2}-\frac{1}{2550}\)

\(\Rightarrow2A=\frac{1275}{2550}-\frac{1}{2550}\)

\(\Rightarrow2A=\frac{637}{1275}\Rightarrow A=\frac{637}{1275}:2=\frac{637}{2550}\)

Vậy tổng dãy phân số trên là :\(\frac{637}{2550}\)

Chúc bạn học tốt !!! :D

\(\frac{4}{7}x+\left(\frac{1}{1.2.3}+\frac{1}{2.3.4}+...+\frac{1}{12.13.14}\right)=\frac{39}{40}\)

\(\Leftrightarrow\frac{4}{7}x+\left[\frac{1}{2}.\left(\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+...+\frac{1}{12.13}-\frac{1}{13.14}\right)\right]=\frac{39}{40}\)

\(\Leftrightarrow\frac{4}{7}x+\left[\frac{1}{2}.\left(\frac{1}{1.2}-\frac{1}{13.14}\right)\right]=\frac{39}{40}\)

\(\Leftrightarrow\frac{4}{7}x+\left[\frac{1}{2}.\frac{45}{91}\right]=\frac{39}{40}\)

\(\Leftrightarrow\frac{4}{7}x+\frac{45}{182}=\frac{39}{40}\)

\(\Leftrightarrow\frac{4}{7}x=\frac{39}{40}-\frac{45}{182}\Leftrightarrow\frac{4}{7}x=\frac{2649}{3640}\)

\(\Rightarrow x=\frac{2649}{3640}\div\frac{4}{7}=\frac{2649}{2080}\)

Vậy x = \(\frac{2649}{2080}\)

đặt A=1.2.3+2.3.4+3.4.5+...+98.99.100

A.4=1.2.3.4+2.3.4.4+3.4.5.4+..+98.99.100.4

A.4=1.2.3(4-0)+2.3.4.(5-1)+3.4.5.(6-2)+......+98.99.100.(101-97)

a.4=1.2.3.4-0+2.3.4.5-2.3.4+3.4.5.6-2.3.4.5+...+98.99.100.101-97.98.99.100

A.4=98.99.100.101

A.4=97990200

A= 97990200:4

A=            24497550

Phải là phân số \(\frac{1}{1.2.3}+\frac{1}{2.3.4}+...+\frac{1}{98.99.100}\) chứ !

A=1999/2000

B=199/200

C=511/512

hok tốt

Đáp án

mình lười trình bày cách làm lém, để đáp án thui nha

A = \(\frac{1999}{2000}\)

B = \(\frac{199}{200}\)

C = \(\frac{511}{512}\)

1/6 + 1/12 + 1/20 + 1/30 + 1/42 + ... + 1/90 + 1/110 = 1/2.3 + 1/3.4 + 1/4.5 + 1/5.6 + 1/6.7 + ... + 1/9.10 + 1/10.11 = 1/2 - 1/3 + 1/3 - 1/4 + 1/4 - 1/5 + 1/5 - 1/6 + 1/6 - 1/7 + ... + 1/9 - 1/10 + 1/10 - 1/11 = 1/2 - 1/11 = 9/22

\(\frac{1}{6}+\frac{1}{12}+\frac{1}{20}+...+\frac{1}{90}+\frac{1}{110}\)

=\(\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+...+\frac{1}{9.10}+\frac{1}{10.11}\)

=\(\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+...+\frac{1}{9}-\frac{1}{10}+\frac{1}{10}-\frac{1}{11}\)

=\(\frac{1}{2}-\frac{1}{11}\)

=\(\frac{9}{22}\)

\(\frac{1}{2}+\frac{1}{6}+...+\frac{1}{90}=\frac{1}{1\cdot2}+\frac{1}{2\cdot3}+...+\frac{1}{9\cdot10}\)

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

                                      \(=\frac{1}{1}-\frac{1}{10}=1-\frac{1}{10}=\frac{9}{10}\)

\(=\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+\frac{1}{5.6}+\frac{1}{6.7}+\frac{1}{7.8}+\frac{1}{8.9}+\frac{1}{9.10}\)

\(=\frac{1}{1}-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{4}-\frac{1}{5}+\frac{1}{6}-\frac{1}{7}+\frac{1}{8}-\frac{1}{9}+\frac{1}{9}-\frac{1}{10}\)

\(=\frac{1}{1}-\frac{1}{10}=\frac{9}{10}\)

Cách 1:

B=1/2+1/4+1/8+1/16+1/32+1/64

B=1-1/2 + 1/2-1/4 + 1/4-1/8 +1/8-1/16 + 1/16-1/32 + 1/32-1/64

B=1-1/64

B=63/64

Cách 2:

B=1/2+1/4+1/8+1/16+1/32+1/64

B=1/2¹+1/2²+1/2³+1/2⁴+1/2⁵+1/2⁶

2B=1+1/2¹+1/2^2+1/2^3+1/2^4+1/2^5

2B-B=1-1/2^6

B=1-1/64 

B=63/64

Đặt A = 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64

2A = 1 + 1/2 + 1/4 + 1/8 + 1/16 + 1/32

2A - A = (1 + 1/2 + 1/4 + 1/8 + 1/16 + 1/32) - (1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64)

A = 1 - 1/64

A = 63/64

Me too