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.