数列{an}中,a1=1/2,an=an-1+1/n2+n(n≥2,n∈N+)则通项公式an

题目

数列{an}中,a1=1/2,an=an-1+1/n2+n(n≥2,n∈N+)则通项公式an
如题

答案

an=a(n-1)+ (1/(n^2+n))
an - a(n-1) = 1/n -1/(n+1)
an -a1 = [1/n -1/(n+1)]+ [1/(n-1) -1/n] +..+ [1/2-1/3]
= 1/2 -1/(n+1)
an= 1-1/(n+1)
= n/(n+1)