一个公差不为零的等差数列An其前n项和Sn,S1 S2 S4为等比数列求A2+A3/A1
问题描述:
一个公差不为零的等差数列An其前n项和Sn,S1 S2 S4为等比数列求A2+A3/A1
答
an = a1+(n-1)d
S1, S2, S4为等比数列
S1.S4 = (S2)^2
a1. [2(2a1+3d)] = (2a1+d)^2
2a1d -d^2 =0
d=2a1
(a2+a3)/a1
=(2a1+3d)/a1
=(2a1+6a1)/a1
=8
答
设公差为d
由题意,
S1=a1
S2=2a1+d
S4=4a1+6d
a1*(4a1+6d)=(2a1+d)^2
d^2-2a1*d=0
d(d-2a1)=0
d不等于0
所以d=2a1
故有(a2+a3)/a1=(a1+d+a1+2d)/a1=(a1+2a1+a1+4a1)/a1=8