高一数列题```要过程

来源：百度知道 编辑：UC知道 时间：2024/06/21 17:34:24
已知等差数列{An}的公差d不等于0，且A1，A3，A9成等比数列，则（A1+A3+A9）/（A2+A4+A10）=多少？

答案是13/16

a3=a1+2d
q9=a1+8d
a1,a3,a9成等比数列
所以(a1+2d)^2=a1*(a1+8d)
a1^2+4a1d+4d^2=a1^2+8a1d
d^2=a1d
d≠0
d=a1
所以a1+a3+a9=a1+(a1+2d)+(a1+8d)=a1+3a1+9a1=13a1
a2+a4+a10=(a1+d)+(a1+3d)+(a1+9d)=2a1+4a1+10a1=16a1
所以(a1+a3+a9)/(a2+a4+a10)=13/16

解:设an的公差是d
∴a3=a1+2d,a9=a1+8d
a2=a1+d,a4=a1+3d,a10=a1+9d
∴a1+a3+a9=3a1+10d,a2+a4+a10=3a1+13d
∵a1,a3,a9依次成等比数列
∴a3/a1=a9/a3
∴a1^2+4d^2+4a1d=a1^2+8a1d
∴a1=d
∴(a1+a3+a9)/(a2+a4+a10)=(3a1+10d)/(3a1+13d)=13d/16d=13/16

A3*A3=A1*A9
(A1+2D)*(A1+2D)=A1*(A1+8D)
D=A1

(A1+A3+A9)/(A2+A4+A10)
=(A1+A1+2D+A1+8D)/(A1+D+A1+3D+A1+9D)
=(3A1+10D)/(3A1+13D)
=13/16