\(\dfrac{1}{4}+\dfrac{1}{3}\times X=\dfrac{1}{2}\\ \dfrac{1}{3}\times X=\dfrac{1}{2}-\dfrac{1}{4}=\dfrac{1}{4}\\ X=\dfrac{1}{4}:\dfrac{1}{3}=\dfrac{1}{4}\times3=\dfrac{3}{4}\)

14+13×�=1213×�=12−14=14�=14:13=14×3=3441​+31​×X=21​31​×X=21​−41​=41​X=41​:31​=41​×3=43

\(\dfrac{3}{4}\)

\(\dfrac{3}{4}\times X-1=\dfrac{1}{2}\times X\\ \dfrac{3}{4}\times X-\dfrac{1}{2}\times X=1\\ \left(\dfrac{3}{4}-\dfrac{1}{2}\right)\times X=1\\ \dfrac{1}{4}\times X=1\\ X=1:\dfrac{1}{4}\\ X=4\)

\(\dfrac{3}{4}\times x-1=\dfrac{1}{2}\times x\\ \dfrac{3}{4}\times x-\dfrac{1}{2}\times x=1\\ x\times\left(\dfrac{3}{4}-\dfrac{1}{2}\right)=1\\ x\times\dfrac{1}{4}=1\\ x=1:\dfrac{1}{4}\\x=4\)

x = 4

D=1/1.5+1/5.9+...+1/41.45

4D=4/1.5+4/5.9+...+4/41.45

4D=1-1/5+1/5-1/9+...+1/41-1/45

4D=1-1/45

D=44/45:4=11/45

Gọi tổng đó là tổng S

Ta có: S = 1/6+1/12+1/30+1/42+1/56+1/72+1/90

=>      S = 1/2.3+1/3.4+1/4.5+1/5.6+1/6.7+1/7.8+1/8.9+1/9.10

=>      S = 1/2-1/3+1/3-1/4+1/4-1/5+1/5-1/6+1/6-1/7+1/7-1/8+1/8-1/9+1/9-1/10

=>      S = 1/2-1/10

=>      S = 5/10-1/10

=>      S=4/10

=>      S=2/5 

Phân số là \(\frac{2}{5}\)

ta thấy: 1/6=1/2x3; 1/12=1/3x4;1/20=1/4x5;.............;1/90=9x10

Đặt A= 1/6+1/12+1/20+...+1/90

A=1/2x3+1/3x4+1/4x5+.....+1/9x10

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

A=1/2-(1/3+1/3)-(1/4+1/4)-......-(1/9+1/9)-1/10

A=1/2-0-0-0-.....0-1/10

A=1/2-1/10

A=2/5

\(B=\frac{6}{1\cdot3}+\frac{6}{3\cdot5}+\cdot\cdot\cdot+\frac{6}{97\cdot99}\)

\(\Rightarrow B=3\cdot\left(\frac{2}{1\cdot3}+\cdot\cdot\cdot+\frac{2}{97\cdot99}\right)\)

\(\Rightarrow B=3\cdot\left(1-\frac{1}{3}+\cdot\cdot\cdot+\frac{1}{97}-\frac{1}{99}\right)\)

\(\Rightarrow B=3\cdot\left(1-\frac{1}{99}\right)\)

\(\Rightarrow B=3\cdot\frac{98}{99}\)

\(\Rightarrow B=\frac{98}{33}\)

\(A=\frac{1}{2}+\frac{1}{6}+\cdot\cdot\cdot+\frac{1}{42}\)

\(\Rightarrow A=\frac{1}{1\cdot2}+\frac{1}{2\cdot3}+\cdot\cdot\cdot+\frac{1}{6\cdot7}\)

\(\Rightarrow A=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\cdot\cdot\cdot+\frac{1}{6}-\frac{1}{7}\)

\(\Rightarrow A=1-\frac{1}{7}\)

\(\Rightarrow A=\frac{6}{7}\)

\(D=\frac{1}{2}+\frac{1}{4}+\cdot\cdot\cdot+\frac{1}{128}\)

\(\Rightarrow2D=1+\frac{1}{2}+\cdot\cdot\cdot+\frac{1}{64}\)

\(\Rightarrow2D-D=\left(1+\cdot\cdot\cdot+\frac{1}{64}\right)-\left(\frac{1}{2}+\cdot\cdot\cdot+\frac{1}{128}\right)\)

\(\Rightarrow D=1-\frac{1}{128}\)

\(\Rightarrow D=\frac{127}{128}\)

\(\dfrac{1}{2}+\dfrac{1}{6}+\dfrac{1}{12}+\dfrac{1}{20}+\dfrac{1}{30}+\dfrac{1}{42}+\dfrac{1}{56}\)

\(=\dfrac{1}{182}\)

1/2 + 1/6 + 1/12 + 1/20 + 1/30 + 1/42 + 1/56 = 1/182

1/182

Có công thức \(\dfrac{x}{a\left(a+x\right)}=\dfrac{1}{a}-\dfrac{1}{a+x}\) nhé!

Ví dụ: \(\dfrac{2}{2.4}=\dfrac{1}{2}-\dfrac{1}{4}\)

\(\dfrac{1}{2}+\dfrac{1}{6}+\dfrac{1}{12}+\dfrac{1}{20}+\dfrac{1}{30}+\dfrac{1}{42}+\dfrac{1}{56}\)

\(=\dfrac{1}{1.2}+\dfrac{1}{2.3}+\dfrac{1}{3.4}+\dfrac{1}{4.5}+\dfrac{1}{5.6}+\dfrac{1}{6.7}+\dfrac{1}{7.8}\)

\(=1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+...+\dfrac{1}{7}-\dfrac{1}{8}\)

\(=1-\dfrac{1}{8}=\dfrac{7}{8}\)

Dấu . tức là nhân nhé!

a) \(=2\sqrt{5}-3\sqrt{5}+\sqrt{5}-1=-1\)

b) \(=\left[\sqrt{14}+\dfrac{\sqrt{6}\left(\sqrt{2}+\sqrt{5}\right)}{\sqrt{2}+\sqrt{5}}\right].\sqrt{\left(\sqrt{\dfrac{7}{2}}-\sqrt{\dfrac{3}{2}}\right)^2}\)

\(=\left(\sqrt{14}+\sqrt{6}\right)\left(\sqrt{\dfrac{7}{2}}-\sqrt{\dfrac{3}{2}}\right)\)

\(=\sqrt{49}-\sqrt{21}+\sqrt{21}-\sqrt{9}\)

\(=7-3=4\)

a) \(\sqrt{20}-15\sqrt{\dfrac{1}{5}}+\sqrt{\left(1-\sqrt{5}\right)^2}=2\sqrt{5}-3\sqrt{5}+\sqrt{5}-1=-1\)

b) \(\left(\dfrac{14}{\sqrt{14}}+\dfrac{\sqrt{12}+\sqrt{30}}{\sqrt{2}+\sqrt{5}}\right).\sqrt{5-\sqrt{21}}=\left(\sqrt{14}+\sqrt{6}\right)\sqrt{5-\sqrt{21}}=\left(\sqrt{7}+\sqrt{3}\right)\sqrt{10-2\sqrt{21}}=\left(\sqrt{7}+\sqrt{3}\right)\left(\sqrt{7}-\sqrt{3}\right)=4\)