题目

求实数a,使下面等式对一切正整数都成立,并用数学归纳法证明你的结论 1/(1×2×3)+1/(2×3×4)+.+1/(n×n+1×n+2)=(n2+an)/(4×n+1×n+2)
都写下..大佬....完了再追100哈

提问时间：2021-03-31

答案

用累加法求和：
1/n(n+1)(n+2)
=1/2[2/n(n+1)(n+2)]
=1/2[(n+2)-n]/n(n+1)(n+2)]
=(1/2)[(n+2)/n(n+1)(n+2)-n/n(n+1)(n+2)]
=(1/2)[1/n(n+1)-1/(n+1)(n+2)]
所以和=(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)]
=n(n+3)/[4(n+1)(n+2)]
数学归纳法的证明我想你自己就不会有问题了吧