正整数列{an},{bn}满足对任意正整数n,an、bn、an+1成等差数列,bn、an+1、bn+1成等比数列,证明：数列{根号bn}成等差数列

Question

正整数列{an},{bn}满足对任意正整数n,an、bn、an+1成等差数列,bn、an+1、bn+1成等比数列,证明：数列{根号bn}成等差数列

Answer 1

当n>=2时,因为bn、an+1、bn+1成等比数列且都是正整数,所以an+1=(bn)^(1/2)*(bn+1)^(1/2),
an=(bn-1)^(1/2)*(bn)^(1/2),
an、bn、an+1成等差数列,所以an+an+1=2bn,把上面的an,an+1代入
(bn)^(1/2)*(bn+1)^(1/2)+(bn-1)^(1/2)*(bn)^(1/2)=2bn
化简得到
根号bn+1+根号bn-1=2倍根号bn
所以数列{根号bn}成等差数列