Question

已知数列{an}满足a1=1,an+1=an+1/n(n+1),则an=

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

Answer 2

题不对啊~~是an+1还是a(n+1),如果是an+1=an+1/n(n+1)，两边减掉an，1=1/n(n+1)。。。不成立吧

Answer 3

a_(n+1)+(1/(n+1))=a_n+1/n 所以a_n+1/n=⋯=a_1+1，∴a_n=2-1/n

Answer 4

an+1 = an + 1/n - 1/(n+1)
an = an-1 + 1/(n -1) - 1/n
an+1 = an-1 + 1/(n-1) - 1/(n+1)=.....= a1 + 1/1 - 1/(n+1) = 2 - 1/(n+1)
an = 2 - 1/n

Answer 5

(an+1)-an=1/n-1/(n+1)
an-a(n-1)=1/(n-1)-1/n
:
:
a2-a1=1-1/2
累加可得(an+1)-a1=1-1/(n+1)
(an+1)=2-1/(n+1)
an=2-1/n