已知数列｛an｝的前n项和Sn=an^2+bn+c(n∈N*），写出｛an｝是等差数列的充要条件

a(1) = S(1) = a + b + c.

n > 1,
a(n) = S(n) - S(n-1) = an^2 + bn + c - a(n-1)^2 - b(n-1) - c

= 2na + b - a

a(n+1) - a(n) = 2a, n = 2,3,...

若｛an｝是等差数列，
则，
a(2) - a(1) = 4a + b - a - a - b - c = 2a - c = 2a,
c = 0.

若
c = 0,
则，
a(2) - a(1) = 2a - c = 2a = a(n+1) - a(n), n = 2,3,...
｛an｝是等差数列.

因此
c = 0是｛an｝是等差数列的充要条件。