Question

等比数列{an}中,an大于0,n属于正整数,求证{lgan}是等差数列,{根号下an}是等比数列

Answer 1

解析：∵{an}是等比数列,令首项=a1,公比=q∴an=a1*q^(n-1),an＞0,n∈N+

∵log[an]-log[a(n-1)]

=log[a1*q^(n-1)]-log[a1*q^(n-2)]

=log{[a1*q^(n-1)]/[a1*q^(n-2)]}

=logq=定值

∴{logan}是首项loga1,公差=logq的等差数列

∵√an/√a(n-1)=√an/a(n-1)=√q=定值

∴{√an}是首项=√a1,公比=√q的等比数列