已知数列{an}、{bn}是等差数列.求证:{pan+qbn}是等差数列.

问题描述:

已知数列{an}、{bn}是等差数列.求证:{pan+qbn}是等差数列.

证明:设数列{an}、{bn}的公差分别为d,d′,则
(pan+1+qbn+1)-(pan+qbn)=p(an+1-an)+q(bn+1-bn)=pd+qd′为常数
∴{pan+qbn}是等差数列.
答案解析:设数列{an}、{bn}的公差,利用等差数列的定义,证明(pan+1+qbn+1)-(pan+qbn)为常数即可.
考试点:等差关系的确定.
知识点:本题考查等差数列的证明,正确运用等差数列的定义是关键.