證明：
當k=1時
1/2+1/3+1/4=13/12=26/24>25/24
結(jié)論成立.
假設k=n時結(jié)論成立,即
1/(n+1)+1/(n+2)+1/(n+3)+...+1/(3n+1)>25/24
當k=n+1時
由于
9(n+1)^2=9n^2+18n+9>9n^2+18n+8=(3n+2)(3n+4)
即
9(n+1)^2/[(3n+2)(3n+4)]-1>0
左側(cè)為
1/[(n+1)+1]+1/[(n+1)+2]+1/[(n+1)+3]+...+1/[3(n+1)+1]
=1/(n+1)+1/(n+2)+1/(n+3)+...+1/(3n+1)+{1/(3n+2)+1/(3n+3)+1/(3n+4)-1/(n+1)}
=1/(n+1)+1/(n+2)+1/(n+3)+...+1/(3n+1)+{6(n+1)/[(3n+2)(3n+4)]-2/(3n+3)}
=1/(n+1)+1/(n+2)+1/(n+3)+...+1/(3n+1)+2/(3n+3)*{9(n+1)^2/[(3n+2)(3n+4)]-1}
>1/(n+1)+1/(n+2)+1/(n+3)+...+1/(3n+1)>25/24.
結(jié)論成立.