等比数列中公比是整数,a1+a4=18,a2+a3=12求前8项之和S8

Question

等比数列中公比是整数,a1+a4=18,a2+a3=12求前8项之和S8

Answer 1

a1+a4=18
a1+a1q^3=18
a1(1+q^3)=18
a1=18/(1+q^3)
[q≠-1]
a2+a3=12
a1q+a1q^2=12
a1(q+q^2)=12
a1=12/(q+q^2)
[q≠-1]
18/(1+q^3)=12/(q+q^2)
3q+3q^2=2+2q^3
2(1+q^3)-3(q+q^2)=0
2(1+q)(1-q+q^2)-3q(1+q)=0
(1+q)(2-2q+2q^2-3q)=0
(1+q)(2q^2-5q+2)=0
(1+q)(q-2)(2q-1)=0
q是整数
所以q-2=0 q=2
a1=12/(q+q^2)=12/6=2
S8=a1*(1-q^8)/(1-q)=2*(1-2^8)/(1-2)=2*127=254