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1/2*4+1/4*6+1/6*8+...+1/48+50=?

先提取公因式。 =(1/4)*(1/1*2+1/2*3+1/3*4+……+1/24*25) =(1/4)*(1/1-1/2+1/2-1/3+1/3-1/4+……+1/24-1/25) =(1/4)*(1-1/25) =6/25

1/2*4+1/4*6+1/6*8+....1/48*50 =1/2*(1/2-1/4)+1/2*(1/4-1/6)+1/2*(1/6-1/8)+......+1/2*(1/48-1/50) =1/2*(1/2-1/4+1/4-1/6+1/6-1/8+......+1/48-1/50) =1/2*(1/2-1/50) =1/2*12/25 =6/25

1/2*4+1/4*6+1/6*8+....1/48*50 =1/2*(1/2-1/4)+1/2*(1/4-1/6)+1/2*(1/6-1/8)+......+1/2*(1/48-1/50) =1/2*(1/2-1/4+1/4-1/6+1/6-1/8+......+1/48-1/50) =1/2*(1/2-1/50) =1/2*12/25 =6/25

2*4/1+4*6/1+6*8/1+.....2014*2016/1 =2分之1×(2分之1-4分之1+4分之1-6分之1+.....+2014分之1-2016分之1) =2分之1×(2分之1-2016分之1) =2分之1×2016分之1007 =4032分之1007

1/4(1/2+1/2*3+1*3*4+.......+1/1007*1008)=1/4(1-1/2+1/2-1/3+1/3-1/4+.......+1/1007-1/1008)=1/4*(1-1/1008)=1007/4032

an = 1/[4n(n+1)] =(1/4)[ 1/n -1/(n+1) ] Sn = a1+a2+...+an =(1/4)[ 1 -1/(n+1) ] = n/[4(n+1)] 1/(2*4)+1/(4*6)+1/(6*8)+...+1/(2014*2016) =S1007 =1007/[4(1007+1)] =1007/4032

1/x-1/(x十2) =[(x十2)-x]/[x(x十1)] =2/ [x(x十1)]

=1/2*(1/2-1/4+1/4-1/6+1/6-1/8+……+1/2012-1/2014) =1/2*(1/2-1/2014) =503/2014

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