Question

已知a1,a2,a3,a4成等比数列,且a1=a2+36,a3=a4+4,求a1,a2,a3,a4?

Answer 1

设公比是q
a2=a1·q
a3=a1·q²
a4=a1·q³
a1-a2=a1-a1·q
=a1(1-q)
=36
a3-a4=a1·q²-a1·q³
=a1·q²·(1-q)
=4
(a3-a4)/(a1-a2)=q²=1/9
q=±1/3
当q=1/3时，(1-1/3)a1=36
a1=54
则a2=18，a3=6，a4=2
当q=-1/3时，[1-(-1/3)]a1=36
a1=27
则a2=-9，a3=3，a4=-1
终上所述：
a1、a2、a3、a4的值为：a1=54，a2=18，a3=6，a4=2
或：a1=27，a2=-8，a3=3，a4=-1

祝您学习愉快

Answer 2

设公比q

a1-a2=36
a3-a4=q^2*(a1-a2)=4

两式相除，得到 q^2=1/9
q=-1/3 或1/3

q=-1/3时:
a1-a2=a1(1-q)=a1*(4/3)=36
a1=27
a2=a1*q=27*(-1/3)=-9
a3=a1*q^2=27*(1/9)=3
a4=a1*q^3=27*(-1/27)=-1

q=1/3时:
a1-a2=a1(1-q)=a1*(2/3)=36
a1=54
a2=a1*q=54*(1/3)=18
a3=a1*q^2=54*(1/9)=6
a4=a1*q^3=54*(1/27)=2

Answer 3

a1 = a2 + 36
a3 = a4 + 4
设公比为k，a2 = ka1, a3 = k*k*a1, a4=k*k*k*a1
所以，
a1(1-k) = 36
a1*k*k*(1-k) = 4
k*k = 1/9
k = 1/3
所以，a1 = 54, a2 = 18, a3 = 6,a4 = 2