Sn=pn^2＋qn＋r,(p,q,r为常数,P不等于零）判断数列是否为等差数列?写出证明过程

Question

Sn=pn^2＋qn＋r,(p,q,r为常数,P不等于零）判断数列是否为等差数列?写出证明过程

Answer 1

S1=a1=p+q+r
an=Sn-S(n-1)=pn^2＋qn＋r-(p(n-1)^2＋q(n-1)＋r)=2pn+q (n>=2)
n>=2时,a(n+1)-an=2p(n+1)+q-(2pn+q )=2p 等差
n=1时,a2-a1=4p+q-(p+q+r)=3p-r
∴若3p-r=2p即r=p时,等差,若r≠p则不等差