Question

已知数列{an}满足:a1=1，an+1=an+1/n(n+1)，则a2013=？

Answer 1

a(n+1)=an +1/[n(n+1)]=an +1/n -1/(n+1)
a(n+1) +1/(n+1)=an +1/n
a1 +1/1=1+1=2
数列{an +1/n}是各项均为2的常数数列。
an +1/n=2
an=2-1/n=(2n-1)/n
a2013=(2×2013-1)/2013=4025/2013

Answer 2

a(n+1)-an=1/n(n+1) an-a(n-1)=1/n(n-1)
a2-a1=1/1x2
a3-a2=1/2x3
a4-a3=1/3x4
---------------
an-a(n-1)=1/n(n-1)
an-a1=1/1x2+1/2x3+1/3x4+---+1/nX(n-1)=1-1/2+1/2-1/3+1/3-1/4+---+1/(n-1)-1/n
=1-1/n
an=a1+1-1/n=2-1/n
a2013=2-1/2013=4025/2013

Answer 3

4025/2013