Question

当n>1且n属于N时,证明1/(n+1) +1/(n+2) +.+1/3n > 9/10 用数学归纳法证明

Answer 1

(1)n=2时,1/(2+1)+1/(2+2)+1/(2+3)+1/(6)=19/20>9/10
(2)假设n=k时不等式成立,即1/(n+1) +1/(n+2) +.+1/3n > 9/10那么当n=k+1时1/(n+2) +.+1/3n +1/(3n+1)+1/(3n+2)+1/(3n+3)
=1/(n+1) +1/(n+2) +.+1/3n+ 1/(3n+1)+1/(3n+2)+1/(3n+3)-1/(n+1)
> 9/10+ 1/(3n+1)+1/(3n+2)+1/(3n+3)-1/(n+1)
因为1/(3n+1)+1/(3n+2)+1/(3n+3)-1/(n+1)>0
所以1/(n+2) +.+1/3n +1/(3n+1)+1/(3n+2)+1/(3n+3)
=1/(n+1) +1/(n+2) +.+1/3n+ 1/(3n+1)+1/(3n+2)+1/(3n+3)-1/(n+1)
> 9/10+ 1/(3n+1)+1/(3n+2)+1/(3n+3)-1/(n+1)
>9/10
综上（1）（2）得证