已知数列{an}的通项公式为an=pn2+qn. (1)当p,q满足什么条件时,数列{an}是等差数列; (2)求证:对任意实数p、q,数列{an+1-an}是等差数列.

问题描述:

已知数列{an}的通项公式为an=pn2+qn.
(1)当p,q满足什么条件时,数列{an}是等差数列;
(2)求证:对任意实数p、q,数列{an+1-an}是等差数列.

(1)∵an=pn2+qn.∴若数列{an}是等差数列;则当n>1时,an-an-1=pn2+qn-[p(n-1)2+q(n-1)]=2pn+q-p为常数,∴必有p=0,即当p=0,数列{an}是等差数列;(2)∵an=pn2+qn.∴当n>1时,an-an-1=pn2+qn-[p(n-1)2...