已知各项都为正数的等比数列,{an}的公比q≠1,且a4,a6,a7成等差数列,则a4+a6a5+a7的值等于:(  ) A.5−12 B.5+12 C.12 D.2

问题描述:

已知各项都为正数的等比数列,{an}的公比q≠1,且a4,a6,a7成等差数列,则

a4+a6
a5+a7
的值等于:(  )
A.
5
−1
2

B.
5
+1
2

C.
1
2

D. 2

设a4=m,公比为q,所以a6=mq2,a7=mq3
a4+a7=2a6
m+mq3=2mq2
1+q3=2q2
(q-1)(q2-q-1)=0∵q≠1
∴q2-q-1=0∴q=

1+
5
2
1−
5
2
(舍)
a4+a6
a5+a7
=
1
q
=
2
1+
5
=
5
−1
2

故选A.