19) \(\sqrt{19-x}=19\)

\(\Rightarrow\sqrt{19-x}=\sqrt{19^2}\)

\(\Rightarrow19-x=19^2\)

\(\Rightarrow19-19^2=x\)

\(\Rightarrow x=19\left(1-19\right)=-19.18=-342\)

21) \(\sqrt{x-1}=\dfrac{1}{3}\)

\(\Rightarrow\sqrt{x-1}=\sqrt{\left(\dfrac{1}{3}\right)^2}\)

\(\Rightarrow x-1=\dfrac{1}{3^2}\)

\(x=\dfrac{1+9}{9}=\dfrac{10}{9}\)

24)\(\sqrt{2x+\dfrac{5}{4}}=\dfrac{3}{2}\)

\(\Rightarrow\sqrt{2x+\dfrac{5}{4}}=\sqrt{\left(\dfrac{3}{2}\right)^2}\)

\(\Rightarrow2x+\dfrac{5}{4}=\left(\dfrac{3}{2}\right)^2=\dfrac{9}{4}\)

\(\Rightarrow2x=\dfrac{9-5}{4}=1\)

\(\Rightarrow x=0,5\)

25) \(\sqrt{\dfrac{x}{3}-\dfrac{7}{6}}=\dfrac{1}{6}\)

\(\Rightarrow\sqrt{\dfrac{2x-7}{6}}=\sqrt{\left(\dfrac{1}{6}\right)^2}\)

\(\Rightarrow\dfrac{2x-7}{6}=\left(\dfrac{1}{6}\right)^2=\dfrac{1}{36}\)

\(\Rightarrow\dfrac{12x-42}{36}=\dfrac{1}{36}\)

\(\Rightarrow12x-42=1\)

\(\Rightarrow12x=43\)

\(\Rightarrow x=\dfrac{43}{12}\)

Ta có A<1/1.2+1/2.3+1/3.4+....+1/19.20

A<1-1/2=1/2-1/3+..+1/19-1/20

A<1-1/20=19/20

Ta có 19/20<19/22(so sánh 2 phân số cùng tử)=>A<19/22  (1)

Ta có A>1/2.3+1/3.4+...+1/20.21

A>1/2-1/3+1/3-1/4+........+1/20-1/21

A>1/2-1/21=20/42

Ta có 20/42>19/42(so sánh 2 phân số cùng mẫu)=>A>19/42  (2)

Từ (1) và (2) =>19/42<A<19/22

\(A=\frac{1}{19}+\frac{2}{19^2}+...+\frac{2014}{19^{2014}}\)

\(\Rightarrow19A=1+\frac{2}{19}+\frac{3}{19^2}+...+\frac{2014}{19^{2013}}\)

\(\Rightarrow19A-A=\left(1+\frac{2}{19}+\frac{3}{19^2}+...+\frac{2014}{19^{2013}}\right)-\left(\frac{1}{19}+\frac{2}{19^2}+...+\frac{2014}{19^{2014}}\right)\)

\(\Rightarrow18A=1+\left(\frac{1}{19}+\frac{1}{19^2}+...+\frac{1}{19^{2013}}\right)-\frac{2014}{19^{2014}}\)

\(\Rightarrow18A=1+\frac{1-\frac{1}{19^{2013}}}{18}-\frac{2014}{19^{2014}}\)

\(\Rightarrow A=\frac{1+\frac{1-\frac{1}{19^{2013}}}{18}-\frac{2014}{19^{2014}}}{18}\)

Vậy...

b) \(\frac{\frac{-6}{5}+\frac{6}{19}-\frac{6}{23}}{\frac{9}{5}-\frac{9}{19}+\frac{9}{23}}=\frac{\left(-6\right).\left(\frac{1}{5}-\frac{1}{19}+\frac{1}{23}\right)}{9.\left(\frac{1}{5}-\frac{1}{19}+\frac{1}{23}\right)}=\frac{-6}{9}=\frac{-2}{3}\)

d) \(\frac{\frac{2}{3}-\frac{2}{5}-\frac{2}{7}+\frac{2}{11}}{\frac{13}{3}-\frac{13}{5}-\frac{13}{7}+\frac{13}{11}}=\frac{2\left(\frac{1}{3}-\frac{1}{5}-\frac{1}{7}+\frac{1}{11}\right)}{13\left(\frac{1}{3}-\frac{1}{5}-\frac{1}{7}+\frac{1}{11}\right)}=\frac{2}{13}\)

Làm tiếp:

\(=\left(1+\frac{1}{2}+.....+\frac{1}{2017}\right)-\left(1+\frac{1}{2}+....+\frac{1}{1008}\right)\)

\(=\frac{1}{1009}+\frac{1}{1010}+.........+\frac{1}{2017}\)

\(\Rightarrow\frac{\frac{1}{1009}+....+\frac{1}{2017}}{1-\frac{1}{2}+.....+\frac{1}{2015}-\frac{1}{2016}+\frac{1}{2017}}=1\)

Bài 2:

Đặt \(A=\frac{1}{2^2}+.......+\frac{1}{2^{800}}\)

\(4A=1+\frac{1}{2^2}+.....+\frac{1}{2^{798}}\)

\(\Rightarrow4A-A=1-\frac{1}{2^{800}}\)

\(\Rightarrow3A=1-\frac{1}{2^{800}}< 1\Rightarrow A< \frac{1}{3}\)

Vậy \(\frac{1}{2^2}+\frac{1}{2^4}+........+\frac{1}{2^{800}}< \frac{1}{3}\)

Bài 1:Tính

a,   Xét biểu thức \(\frac{\left(1+\frac{n}{1}\right)\left(1+\frac{n}{2}\right).........\left(1+\frac{n}{n+2}\right)}{\left(1+\frac{n+2}{1}\right)\left(1+\frac{n+2}{2}\right)..........\left(1+\frac{n+2}{n}\right)}\) với\(n\in N\)

Ta có:\(\frac{\left(1+\frac{n}{1}\right)\left(1+\frac{n}{2}\right).......\left(1+\frac{n}{n+2}\right)}{\left(1+\frac{n+2}{1}\right)\left(1+\frac{n+2}{2}\right)......\left(1+\frac{n+2}{n}\right)}\)

\(=\frac{\frac{n+1}{1}.\frac{n+2}{2}........\frac{2n+2}{n+2}}{\frac{n+3}{1}.\frac{n+4}{2}.........\frac{2n+2}{n}}\)

\(=\frac{\frac{\left(n+1\right)\left(n+2\right).......\left(2n+2\right)}{1.2.3.........\left(n+2\right)}}{\frac{\left(n+3\right)\left(n+4\right)........\left(2n+2\right)}{1.2.3.........n}}\)

\(=\frac{\left(n+1\right)\left(n+2\right).......\left(2n+2\right).1.2.3.......n}{\left(n+3\right)\left(n+4\right)........\left(2n+2\right).1.2.3......\left(n+2\right)}\)

\(=\frac{\left(n+1\right)\left(n+2\right)}{\left(n+1\right)\left(n+2\right)}=1\)

Áp dụng vào bài toán ta có đáp số là:1

b, \(\frac{\frac{-6}{5}+\frac{6}{19}-\frac{6}{23}}{\frac{9}{5}-\frac{9}{19}+\frac{9}{23}}=\frac{\left(-6\right).\left(\frac{1}{5}-\frac{1}{19}+\frac{1}{23}\right)}{9.\left(\frac{1}{5}-\frac{1}{19}+\frac{1}{23}\right)}=\frac{-6}{9}=-\frac{2}{3}\)

c,\(\frac{\frac{1}{6}-\frac{1}{39}+\frac{1}{51}}{\frac{1}{8}-\frac{1}{52}+\frac{1}{68}}=\frac{\frac{1}{3}.\left(\frac{1}{2}-\frac{1}{13}+\frac{1}{17}\right)}{\frac{1}{4}.\left(\frac{1}{2}-\frac{1}{13}+\frac{1}{17}\right)}=\frac{\frac{1}{3}}{\frac{1}{4}}=12\)

d,\(\frac{\frac{2}{3}-\frac{2}{5}-\frac{2}{7}}{\frac{13}{3}-\frac{13}{5}-\frac{13}{7}}=\frac{2\left(\frac{1}{3}-\frac{1}{5}-\frac{1}{7}\right)}{13\left(\frac{1}{3}-\frac{1}{5}-\frac{1}{7}\right)}=\frac{2}{13}\)

e,Xét mẫu số ta có:

\(1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+..........+\frac{1}{2015}-\frac{1}{2016}+\frac{1}{2017}\)

\(=1+\frac{1}{2}-2.\frac{1}{2}+\frac{1}{3}+\frac{1}{4}-2.\frac{1}{4}+.....+\frac{1}{2015}+\frac{1}{2016}-2.\frac{1}{2016}+\frac{1}{2017}\)

\(=\left(1+\frac{1}{2}+\frac{1}{3}+.......+\frac{1}{2017}\right)-2.\left(\frac{1}{2}+\frac{1}{4}+.........+\frac{1}{2016}\right)\)

Sửa đề:\(\frac{\frac{2}{11}-\frac{2}{13}-\frac{2}{19}}{\frac{5}{19}-\frac{5}{11}-\frac{5}{13}}\)

\(=\frac{2.\left(\frac{1}{11}-\frac{1}{13}-\frac{1}{19}\right)}{5.\left(\frac{1}{19}-\frac{1}{11}-\frac{1}{13}\right)}\)

\(=\frac{2}{5}\)

\(\frac{\frac{2}{11}+\frac{2}{13}-\frac{2}{19}}{\frac{5}{19}-\frac{5}{11}-\frac{5}{13}}\)

\(=\frac{\frac{48}{143}-\frac{2}{19}}{\frac{-40}{209}-\frac{5}{13}}\)

\(=\frac{\frac{626}{2717}}{\frac{-1565}{2717}}\)

\(=\frac{626}{-1565}\)

\(=-0,4\)