=1/(1x3)x1/(2x4)x...x1/(9x11)

=1/(1x3x2x4x...x9x11)

=1/(1x2x3x3x4x4x5x5x...x9x10x11)

Ta có: \(P=1\frac13\times1\frac18\times1\frac{1}{15}\times\ldots\times1\frac{1}{80}\times1\frac{1}{99}\)

\(=\frac43\times\frac98\times\frac{16}{15}\times\ldots\times\frac{81}{80}\times\frac{100}{99}\)

\(=\frac{2\times2}{1\times3}\times\frac{3\times3}{2\times4}\times\ldots\times\frac{10\times10}{9\times11}\)

\(=\frac{2\times3\times\ldots\times10}{1\times2\times\ldots\times9}\times\frac{2\times3\times\ldots\times10}{3\times4\times\ldots\times11}=\frac{10}{1}\times\frac{2}{11}=\frac{20}{11}\)

TA có: \(1\frac13\times1\frac18\times\ldots\times1\frac{1}{99}\)

\(=\frac43\times\frac98\times\ldots\times\frac{100}{99}\)

\(=\frac{2\times2}{1\times3}\times\frac{3\times3}{2\times4}\times\ldots\times\frac{10\times10}{9\times11}\)

\(=\frac{2\times3\times\ldots\times10}{1\times2\times\ldots\times9}\times\frac{2\times3\times\ldots\times10}{3\times4\times\ldots\times11}\)

\(=\frac{10}{1}\times\frac{2}{11}=\frac{20}{11}\)

Ta có công thức:

\(1-\frac{3}{n\left(n+2\right)}\)

\(=\frac{n\left(n+2\right)-3}{n\left(n+2\right)}\)

\(=\frac{n^2+2n-3}{n\left(n+2\right)}=\frac{\left(n+3\right)\left(n-1\right)}{n\left(n+2\right)}\)

Ta có: \(A=\left(1-\frac38\right)\times\left(1-\frac{3}{15}\right)\times\ldots\times\left(1-\frac{3}{143}\right)\)

\(=\left(1-\frac{3}{2\times4}\right)\times\left(1-\frac{3}{3\times5}\right)\times\ldots\times\left(1-\frac{3}{11\times13}\right)\)

\(=\frac{\left(2+3\right)\left(2-1\right)}{2\times\left(2+2\right)}\times\frac{\left(3+3\right)\left(3-1\right)}{3\times\left(3+2\right)}\times\ldots\times\frac{\left(11+3\right)\times\left(11-1\right)}{11\times\left(11+2\right)}\)

\(=\frac{5\cdot6\cdot..\cdot14}{4\cdot5\cdot..\cdot13}\cdot\frac{1\cdot2\cdot\ldots\cdot10}{2\cdot3\cdot\ldots\cdot11}=\frac{14}{4}\cdot\frac{1}{11}=\frac72\cdot\frac{1}{11}=\frac{7}{22}\)

\(K=\left(1-\dfrac{3}{2\cdot4}\right)\left(1-\dfrac{3}{3\cdot5}\right)\cdot...\cdot\left(1-\dfrac{3}{19\cdot21}\right)\)

\(=\dfrac{3^2-1-3}{\left(3-1\right)\left(3+1\right)}\cdot\dfrac{4^2-1-3}{\left(4-1\right)\left(4+1\right)}\cdot...\cdot\dfrac{20^2-4}{\left(20-1\right)\left(20+1\right)}\)

\(=\dfrac{\left(3-2\right)\left(3+2\right)}{\left(3-1\right)\left(3+1\right)}\cdot\dfrac{\left(4-2\right)\left(4+2\right)}{\left(4-1\right)\left(4+1\right)}\cdot...\cdot\dfrac{18\cdot22}{\left(20-1\right)\left(20+1\right)}\)

\(=\dfrac{1\cdot5}{2\cdot4}\cdot\dfrac{2\cdot6}{3\cdot5}\cdot...\cdot\dfrac{18\cdot22}{19\cdot21}\)

\(=\dfrac{1\cdot2\cdot3\cdot...\cdot21\cdot22}{2\cdot3\cdot4\cdot5\cdot...\cdot19\cdot20\cdot21}=1\cdot22=22\)

\(...=\dfrac{4}{3}x\dfrac{9}{8}x\dfrac{16}{15}x\dfrac{25}{24}=\dfrac{4x9x16x25}{3x8x15x24}=\dfrac{5}{3}\)

\(=\dfrac{4}{3}\cdot\dfrac{9}{8}\cdot\dfrac{16}{15}\cdot\dfrac{25}{24}\cdot\dfrac{36}{35}=\dfrac{12}{7}\)

= 4/3*9/8*16/15*25/24*36/35

=2*2/1*3 * 3*3/2*4 *4*4/3*5 *5*5/4*6 * 6*6/5*7

= (2*3*4*5*6 / 1*2*3*4*5) * ( 2*3*4*5*6 / 3*4*5*6*7)

=6/1* 2/7 

= 12/7

1/3x1/8x...x1/99

=1/(1x3)x1/(2x4)x...x1/(9x11)

=1/(1x3x2x4x...x9x11)

=1/(1x2x3x3x4x4x5x5x...x9x9x10x11)

tính tích của 10 hỗn số đầu tiên trong dãy các hỗn số sau : 

1 và 1/3 x 1 và 1/8 x 1 và 1/15 x 1 và 1/24 x 1 và 1/35 x .......