数列an的前n项和为Sn=an*2+bn+c,则数列an是等差数列的充要条件是

Sn=an²+bn+c
Sn-1=a(n-1)²+b(n-1)+c (n≥2)
an=Sn-Sn-1=an²-a(n-1)²+b
=(2n-1)a+b
当n=1时
Sn=a+b+c
an是等差数列说明S1=a1满足an=(2n-1)a+b
a+b+c=a+b
c=0