求证:1/1*3+1/2*4+1/3*5+……+1/n(n+2)=3/4-2n+3/2(n+1)(n+2)
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1/1*3+1/2*4+1/3*5+……+1/n(n+2)
=[1-1/3+1/2-1/4+1/3-1/5+1/4-1/6+1/5-1/7+......+1/(n-1)-1/(n-3)+1/n-1/(n+2)]×1/2
=[1+1/2-1/(n+1)-1/(n+2)]×1/2
=[3/2-(2n+3)/(n+1)(n+2)]×1/2
=3/4-(2n+3)/2(n+1)(n+2)
=[1-1/3+1/2-1/4+1/3-1/5+1/4-1/6+1/5-1/7+......+1/(n-1)-1/(n-3)+1/n-1/(n+2)]×1/2
=[1+1/2-1/(n+1)-1/(n+2)]×1/2
=[3/2-(2n+3)/(n+1)(n+2)]×1/2
=3/4-(2n+3)/2(n+1)(n+2)
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