已知{an}是公差不为零的等差数列{bn}为等比数列,满足b1=a1^2,b2=a2^2,b3=a3^2求{bn}公比q的值(n,1,2,3均为下标）
问题描述：
已知{an}是公差不为零的等差数列{bn}为等比数列,满足b1=a1^2,b2=a2^2,b3=a3^2
求{bn}公比q的值
(n,1,2,3均为下标）
答
令{an}公差为d,由b2^2=b1*b3得：
a2^4=a1^2*a3^2
两边开方得：
a2^2=a1*a3或a2^2=-a1*a3
当a2^2=a1*a3时,有：
(a1+d)^2=a1(a1+2d)
化简得d^2=0,即d=0与公差不为零相矛盾,无解.
2.当a2^2=-a1*a3时,有：
(a1+d)^2=-a1(a1+2d)
化简得2a1^2+4a1d+d^2=0
解之得a1=(-1+√2/2)d 或 a1=(-1-√2/2)d
①若a1=(-1+√2/2)d ,则：
q=a2^2/a1^2=(a1+d)^2/a1^2
=[(-1+√2/2)d+d]^2/[(-1+√2/2)d ]^2
=3+2√2
②若a1=(-1-√2/2)d ,则：
q=a2^2/a1^2=(a1+d)^2/a1^2
=[(-1-√2/2)d+d]^2/[(-1-√2/2)d ]^2
=3-2√2
故{bn}公比q的值为3+2√2或3-2√2