哈雷彗星撞地球啦!已知6sn=an^2=3an+2.a1,a3,a11成等比数列,an+an-1不等0,

题目

哈雷彗星撞地球啦!已知6sn=an^2=3an+2.a1,a3,a11成等比数列,an+an-1不等0,
完整的题目是；
已知6sn=an^2+3an+2.a1,a3,a11成等比数列,an+an-1不等于0,求an的通项公式

答案

6Sn - 6S(n-1) = 6an = an^2 + 3an - (a(n-1))^2 - 3a(n-1)
(an - a(n-1) - 3)(an + a(n-1))=0
由于an + a(n-1)不等于0
an - a(n-1) - 3 = 0
an = a1 + 3(n - 1)
n=1时
6S1=6a1=a1^2+3a1+2
a1=1,a1=2
a1=1时,a3=7,a11=31,不符题意
a1=2时,a3=8,a11=32,为等比数列,符合题意
故an = 2 + 3(n - 1) = 3n - 1