Question

a5-a1=15,a4-a2=6,求a3

Answer 1

an是等比数列
a5-a1=15,
a3q^2-a3/q^2=15.................1
a4-a2=6

a3q-a3/q=6...............2
1式除以2式得
(q^2-1/q^2)/(q-1/q)=5/2
2(q^2-1/q^2)=5(q-1/q)
2(q^2-1/q^2)-5(q-1/q)=0
2(q-1/q)(q+1/q)-5(q-1/q)=0
(q-1/q)[2(q+1/q)-5]=0
(q-1/q)(2q+2/q-5)=0
q-1/q=0或2q+2/q-5=0
q^2-1=0或2q^2-5q+2=0
(q-1)(q+1)=0或(2q-1)(q-2)=0
q=1或q=-1(舍去)或q=1/2或q=2

当q=1/2时
a5-a1=15
a1q^4-a1=15
-15a1/16=15
a1=-16
a3=a1q^2
=-16*1/4
=-4

当q=2时
a5-a1=15
a1q^4-a1=15
15a1=15
a1=1
a3=a1q^2
=1*4
=4