设各项均为正数的无穷数列an和bn满足2bn=an+an+1且an-1方=bn*bn+1,求证根号bn是等差数列

题目

设各项均为正数的无穷数列an和bn满足2bn=an+an+1且an-1方=bn*bn+1,求证根号bn是等差数列
a1=1,a2=2求an和bn的通项公式

答案

题意：an+an+1=2bn; (1) bnbn+1=an+1*an+1 (2)（2）式两边开方得：an+1=sqrt(bn)*sqrt(bn+1) (3)(1)式两边平方,展开,然后将（3）式代入,可得：bn*bn-1+bn*bn+1+2*sqrt(bn-1*bn*bn*bn+1)=4bn*bn (4)整理（4）式后可得...