a) \(\frac{1}{2}\times\frac{12}{13}+\frac{1}{3}\times\frac{12}{13}+\frac{1}{4}\times\frac{12}{13}\)

  = (1/2 + 1/3 + 1/4) x 12/13

  = 13/12 x 12/13

  = 1

b) (1 - 1/2 ) x (1 - 1/3) x (1 - 1/4) x ( 1  - 1/5)

=   1/2 x 2/3 x 3/4 x 4/5

=    1/5

ở tử số ta làm thế này

\(TS=\left(1+\frac{1}{2014}\right)+\left(1+\frac{1}{2013}\right)+\left(1+\frac{1}{2012}\right)+...+\left(1+\frac{2013}{2}\right)\)

\(TS=2015\left(\frac{1}{2014}+\frac{1}{2013}+\frac{1}{2012}+...+\frac{1}{2}\right)\)

\(\frac{TS}{MS}=2015\)

chứng minh \(\frac{3}{2}\ge\frac{x}{1+x^2}+\frac{y}{1+y^2}+\frac{z}{1+z^2}\)

ta có \(\left(x-1\right)^2\ge0\Leftrightarrow x^2+1\ge2x\Leftrightarrow\frac{2x}{1+x^2}\le1\)

\(\left(y-1\right)^2\ge0\Leftrightarrow y^2+1\ge2y\Leftrightarrow\frac{2y}{1+y^2}\le1\)

\(\left(z-1\right)^2\ge0\Leftrightarrow z^2+1\ge2z\Leftrightarrow\frac{2z}{1+z^2}\le1\)

\(\Rightarrow\frac{2x}{1+x^2}+\frac{2y}{1+y^2}+\frac{2x}{1+z^2}\le3\Leftrightarrow\frac{x}{1+x^2}+\frac{y}{1+y^2}+\frac{z}{1+z^2}\le\frac{3}{2}\)

chứng minh \(\frac{1}{1+x}+\frac{1}{1+y}+\frac{1}{1+z}\ge\frac{3}{2}\)

áp dụng bất đẳng thức Cauchy ta có: 

\(\frac{1}{1+x}+\frac{1}{1+y}+\frac{1}{1+z}\ge3\sqrt[3]{\frac{1}{\left(1+x\right)\left(1+y\right)\left(1+z\right)}}=\frac{3}{\sqrt{\left(1+x\right)\left(1+y\right)\left(1+z\right)}}\)

ta lại có \(\frac{\left(1+x\right)\left(1+y\right)\left(1+z\right)}{3}\ge\sqrt[3]{\left(1+x\right)\left(1+y\right)\left(1+z\right)}\)

vậy \(\frac{1}{1+x}+\frac{1}{1+y}+\frac{1}{1+z}\ge\frac{3}{\frac{\left(1+x\right)+\left(1+y\right)+\left(1+z\right)}{3}}=\frac{3}{2}\)

kết hợp ta có \(\frac{x}{1+x^2}+\frac{y}{1+y^2}+\frac{z}{1+z^2}\le\frac{3}{2}\le\frac{1}{1+x}+\frac{1}{1+y}+\frac{1}{1+z}\)

M=\(\frac{1}{1\cdot2}\)- \(\frac{1}{2\cdot3}\)+ \(\frac{1}{2\cdot3}\) - \(\frac{1}{3\cdot4}\) +..........+ \(\frac{1}{10\cdot11}\) - \(\frac{1}{11\cdot12}\)

= \(\frac{1}{2}\) - \(\frac{1}{11\cdot12}\) 

=\(\frac{65}{132}\) 

Đây không phải toán lớp 3, em nhé.

1246

co len viet nam

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Tôi yêu Việt Nam

1234+12 = 1246

1+2+3+4+5+6+7+8+9+1+2+3+4+5+6+7+8+9+0+0+0+3+4+1+2+3+111+44+99 = 357

1+2+3+4+5+6+7+8+9+1+2+3+4+5+6+7+8+9+0+0+0+3+4+1+2+3+111+44+99=357

=1x12+2x8+3x6=12+16+18=46

= 46 nha 

nhớ k

kb nữa

kết quả là:

1/362880

k mình nha