Question

等差数列{an}中a1,a3,a7成等比，求公比q

Answer 1

设等差数列公差为d，则a3=a1+2d，a7=a1+6d
又a3/a1=q，即a1+2d=a1·q
a7/a3=q，即a1+6d=(a1+2d)q=a1·q+2qd
两式相减，得4d=2qd，所以公比q=2

Answer 2

an = a1+(n-1)d
a1,a3,a7成等比
a1.a7= (a3)^2
a1(a1+6d) = (a1+2d)^2
(a1)^2+6a1.d = (a1)^2 +4a1.d +4d^2
2a1.d-4d^2=0
d(a1-2d)=0
a1=2d
公比q
= a3/a1
= (a1+2d)/a1
= (a1+a1)/a1
=2