二阶等差数列是从第二项起,每一项与前一项的差构成的数列是等差数列.
{an}是二阶等差数列,则{a(n+1)-an}是等差数列.设a(n+1)-an=pn+q,
an=an-a(n-1)+a(n-1)-a(n-2)+…+a2-a1+a1
=[p(n-1)+q]+[p(n-2)+q]+…+(p+q)+a1
=pn(n-1)/2+q(n-1)+a1
=p/2n^2+(q-p/2)n+a1-q
具有an=pn^2+qn+r的形式,其中p,q,r是任意常数.