已知a1+a2+……+an=n×n×n,求:1/(a2-1)+1/(a3-1)+1/(a4-1)+……+1/(a2011-1)=?
人气:130 ℃ 时间:2020-06-21 03:17:38
解答
∵a1+a2+……+an=n×n×n
∴an=n^3-(n-1)^3=n^2+n*(n+1)+(n+1)^2=n^2+(n^2+n)+(n^2+2n+1)=3n^2+3n+1
1/(an-1)=1/(3n^2+3n+1-1)=1/(3n*(n+1))=1/3*[1/n - 1/(n+1)]
1/(a2-1)+1/(a3-1)+1/(a4-1)+……+1/(a2011-1)
=1/3*{(1/2 - 1/3)+(1/3 - 1/4)+(1/4 - 1/5)+……(1/2011 - 1/2012)}
=1/3*(1/2 - 1/2012)
=1/3*(2012-2)/(2012*2)
=335/2012
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