设数列{an}的前n项和为Sn,a1=1,an=sn/n+2(n-1),求证数列{an}是等差数列,并求其通项公式an_数学解答

设数列{an}的前n项和为Sn,a1=1,an=sn/n+2(n-1),求证数列{an}是等差数列,并求其通项公式an

人气：293 ℃　时间：2019-09-23 14:02:27

解答

an=sn/n+2(n-1)
Sn=nan-2n(n-1)
S(n-1)=(n-1)a(n-1)-2(n-1)(n-2)
Sn-S(n-1)=an=nan-2n(n-1)-[(n-1)a(n-1)-2(n-1)(n-2)]
=nan-2n(n-1)-(n-1)a(n-1)+2(n-1)(n-2)
=nan-2n(n-1)-na(n-1)+a(n-1)+2n(n-1)-4(n-1)
=n[an-a(n-1)]+a(n-1)-4(n-2)
n[an-a(n-1)]-[an-a(n-1)]=4(n-1)
[an-a(n-1)](n-1)=4(n-1)
an-a(n-1)=4
所以数列{an}是等差数列,公差d=4
An=a1+(n-1)d=1+(n-1)*4=4n-3