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\(C=1.2+2.3+3.4+...+x.\left(x-1\right)\)
\(\Rightarrow3C=1.2.3+2.3.3+3.4.3+...+x.\left(x-1\right).3\)
\(\Rightarrow3C=1.2.\left(3-0\right)+2.3.\left(4-1\right)+3.4.\left(5-2\right)+...+x.\left(x-1\right).\left[\left(x+1\right)-\left(x-2\right)\right]\)
\(\Rightarrow3C=\left(1.2.3-0.12\right)+\left(2.3.4-1.2.3\right)+\left(3.4.5-2.3.4\right)+...+\left[x.\left(x-1\right)\left(x+1\right)-x.\left(x-1\right)\left(x-2\right)\right]\)
\(\Rightarrow3C=-0.1.2+x.\left(x-1\right)\left(x+1\right)\)
\(\Rightarrow3C=x.\left(x-1\right)\left(x+1\right)\)
\(\Rightarrow C=\dfrac{x.\left(x-1\right)\left(x+1\right)}{3}\)
3C=1x2x3+2x3x3+3x4x3+...+Xx(X+1)=
=1x2x3+2x3x(4-1)+3x4x(5-2)+...+Xx(X+1)[(X+2)-(X-1)]=
=1x2x3-1x2x3+2x3x4-2x3x4+3x4x5-...-(X-1)xXx(X+1)+Xx(X+1)x(X+2)=
=Xx(X+1)(X+2)
\(A=\frac{1}{2.2}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{49.50}\)
\(A=\frac{1}{2}-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{49}-\frac{1}{50}\)
\(\Rightarrow A=\frac{1}{2}-\frac{1}{50}\)
\(\Rightarrow A=\frac{12}{25}\)
Vậy \(A=\frac{12}{25}\)
P=1x2+2x3+3x4+...+2017x2018
3P = 1x2x3 + 2x3x3 + 3x4x3 + ... + 2017x2018x3
3P = 1x2x3 + 2x3x(4-1) + 3x4x(5-2) + ... +2017x2018x(2019-2016)
3P = 1x2x3 + 2x3x4 - 1x2x3 + 3x4x5 - 2x3x4 + ... + 2017x2018x2019 - 2016x2017x2018
3P = 2017x2018x2019
P = 2017x2018x2019 : 3
P = 2739315938
P = 1x2+2x3+3x4+...+2017x2018
3xP = 1x2x3+2x3x3+3x4x3+...+2017x2018x3
3xP = 1x2x3+2x3x(4-1)+3x4x(5-2)+...+2017x2018x(2019-2016)
3xP = 1x2x3+2x3x4-2x3x1+3x4x5-3x4x2+...+2017x2018x2019-2017x2018x2016
3xP = 2017x2018x2019
3xP = 8217947814
P = 8217947814 : 3
P = 2739315938
1) Đặt A = 1.2 + 2.3 + 3.4 + ...... + 2008.2009
<=> 3A = 1.2.3 + 2.3.3 + 3.4.3 + ...... + 2008.2009.3
<=> 3A = 1.2.3 + 2.3.( 4 - 1 ) + 3.4.( 5 - 2 ) + .... + 2008.2009.( 2010 - 2007 )
<=> 3A = 1.2.3 + 2.3.4 - 1.2.3 + 3.4.5 - 2.3.4 + .... + 2008.2009.2010 - 2007.2008.2009
<=> 3A = 2008.2009.2010
=> A = ( 2008.2009.2010 ) : 3
=3x16-2x9=48-18=30
3.4.4-2.3.3
=6