Question

证明：对于任意的正整数n,有1/1*2*3+1/2*3*4+......+1/n(n+1)(n+2)<1/4

Answer 1

证明：设数列{an}，an=1/n(n+1)(n+2),
则an=1/2{[1/n-1/(n+1)]-[(1/n+1)-1/(n+2)]},
得1/1*2*3+1/2*3*4+......+1/n(n+1)(n+2)
=a1+a2+……+an
=1/2[(1-1/2)-(1/2-1/3)]+1/2[(1/2-1/3)-(1/3-1/4)]+……+1/2{[1/n-1/(n+1)]-[(1/n+1)-1/(n+2)]}
=1/2(1-1/2)-1/2[(1/n+1)-1/(n+2)]
=1/4-1/2[(1/n+1)-1/(n+2)]
由于n+1<n+2，即1/2[(1/n+1)-1/(n+2)]>0，则
1/1*2*3+1/2*3*4+......+1/n(n+1)(n+2)=1/4-1/2[(1/n+1)-1/(n+2)]<1/4

Answer 2

因为1/[n(n+1)(n+2)]=1/2{1/n-1/(n+1)-1/(n+2)}
所以原等式=1/2{1-1/(n+1)-1/(n+2)}所以原不等式成立。

Answer 3

黑糊糊黑糊糊黑糊糊后