For what real numbers $a, b, c$ is it true that if the sum of the first $n$ terms of a sequence is of the form $a n^{2}+b n+c$ for any positive integer $n$, then the sequence is an arithmetic sequence?
Let the elements of the sequence be $a_{1}, a_{2}, \ldots, a_{n}, \ldots$; the sum of the first $n$ elements is $S_{n}$. According to the problem, $S_{n}=a n^{2}+b n+c$. This defines the elements of the sequence:

$$
a_{1}=S_{1}=a+b+c ; a_{n}=S_{n}-S_{n-1}=\left(a n^{2}+b n+c\right)-\left(a(n-1)^{2}+b(n-1)+c\right)=a(2 n-1)+b, \quad \text { if } n \geq 2
$$

The difference between consecutive elements:

$$
a_{2}-a_{1}=(3 a+b)-(a+b+c)=2 a-c ; a_{n}-a_{n-1}=(a(2 n-1)+b)-(a(2 n-3)+b)=2 a, \quad \text { if } n \geq 3
$$

The sequence is exactly an arithmetic sequence if these differences are equal, that is, $c=0$.