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

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

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

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

a) 2/3 : 3/5 × 5/7 : 2/3

= 2/3 × 5/3 × 5/7 × 3/2 

= 25/21

b) 1 1/2 × 1 1/3 × 1 1/18 × 1 1/15 × 1 1/24 × 1 1/35

= 3/2 × 4/3 × 19/18 × 16/15 × 25/24 × 36/35

= 2 × 152/35 × 15/14

= 304/35 × 15/14

= 152/7

a:Sửa đề: 1x3+2x4+...+99x101

\(=1\times\left(1+2\right)+2\times\left(2+2\right)+\cdots+99\times\left(99+2\right)\)

\(=\left(1\times1+2\times2+\cdots+99\times99\right)+2\times\left(1+2+\cdots+99\right)\)

\(=\frac{99\times\left(99+1\right)\times\left(2\times99+1\right)}{6}+2\times\frac{99\times100}{2}\)

\(=\frac{99\times100\times199}{6}+99\times100=33\times50\times199+99\times100\)

\(=33\times50\times\left(199+3\times2\right)=33\times50\times205=338250\)

b: \(\frac89\times\frac{15}{16}\times\ldots\times\frac{2499}{2500}\)

\(=\left(1-\frac19\right)\times\left(1-\frac{1}{16}\right)\times\ldots\times\left(1-\frac{1}{2500}\right)\)

\(=\left(1-\frac13\right)\times\left(1-\frac14\right)\times\ldots\times\left(1-\frac{1}{50}\right)\times\left(1+\frac13\right)\times\left(1+\frac14\right)\times\ldots\times\left(1+\frac{1}{50}\right)\)

\(=\frac23\times\frac34\times\ldots\times\frac{49}{50}\times\frac43\times\frac54\times\ldots\times\frac{51}{50}=\frac{2}{50}\times\frac{51}{3}=\frac{17}{25}\)

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

= 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

ban lam sai rui de mk lam lai nhe.

\(12.\left(x-1\right):3=4^3-2^3\)

\(12.\left(x-1\right):3=64-8\)

\(12.\left(x-1\right):3=56\)

\(12.\left(x-1\right)=56.3\)

\(12.\left(x-1\right)=168\)

\(x-1=168:12\)

\(x-1=14\)

             \(x=15\)

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

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

=1/(1x3x2x4x...x9x11)

=1/(1x2x3x3x4x4x5x5x...x9x9x10x11)