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问题: 数字题

1/2+5/6+11/12+19/20+...+89/90+109/110的值是?

解答:

1/2+5/6+11/12+19/20+...+89/90+109/110
=(1-1/2)+(1-1/6)+(1-1/12)+(1-1/20)+……+(1-1/90)+(1-1/110)
=1*10-(1/2+1/6+1/12+1/20+……+1/90+1/110)
=10-(1/2+1/6+1/12+1/20+……+1/90+1/110)
1/2+1/6+1/12+1/20+……+1/90+1/110=1/(1*2)+1/(2*3)+1/(3*4)+……+1/(9*10)+1/(10*11)
1/[n(n+1)]=[(n+1)-n]/[n(n+1)]=1/n-1/(n+1)
所以1/2+1/6+1/12+1/20+……+1/90+1/110=1/(1*2)+1/(2*3)+1/(3*4)+……+1/(9*10)+1/(10*11)=1/1-1/2+1/2+1/3+……+1/9-1/10+1/10-1/11=1-1/11
所以原式10-1+1/11=9+1/11=100/11