Question

等差数列an中，公差为d（不等于0），且a1,a3,a9成等比数列，则（a1+a3+a9)/(a2+a4+a9)的值为

Answer 1

a3=a1+2d
a9=a1+8d
a1a9=a3^2
a1*(a1+8d)=a1^2+8a1d
(a1+2d)^2=a1^2+4a1d+4d^2
a1^2+8a1d=a1^2+4a1d+4d^2
(a1-d)d=0
因为d不等于0
a1=d
（a1+a3+a9)/(a2+a4+a9)
=(a1+3a1+9a1)/(2a1+4a1+9a1)
=13a1/15a1
=13/15

Answer 2

a3^2=a1*a9
(a1+2d)^2=a1*(a1+8d)
a1^2+4d^2+4da1=a1^2+8da1
4d^2=4da1
4d=4a1
d=a1
a1=d
a2=2d
a3=3d......
∴（a1+a3+a9)/(a2+a4+a9)
=d+3d+9d/2d+4d+9d
=13d/15d
=13/15