Question

求证:对任意的正整数n,有1×2×3分之1+2×3×4分之1+.n(n+1)(n+2)分之1小于1/4

Answer 1

1/1×2×3=1/2×(1/1×2-1/2×3)

其他以此类推

所以左边=1/2×[1/1×2-1/2×3+1/2×3-1/3×4+……+1/n(n+1)-1/(n+1)(n+2)]

=1/2×[1/1×2-1/(n+1)(n+2)]

=1/4-1/2(n+1)(n+2)