Question

等比数列{an}中,Sn表示前n项和,a3=3S2+1,a4=2S3+1,则公比q为

Answer 1

a3=3S2+1,
a3-3S2=1,
a4=2S3+1
a4-2S3=1

a3-3S2=a4-2S3
a3=a4-2S3+3S2
a3=a4-2S3+2S2+s2
a3=a4-2(S3-S2)+s2
a3=a4-2a3+s2
3a3=a4+s2
4a3=a4+s2+a3
4a3=s4
4a1q^2=a1(1-q^4)/(1-q)
4q^2=(1-q^4)/(1-q)
4q^2=(1+q^2)(1+q)(1-q)/(1-q)
4q^2=(1+q^2)(1+q)
4q^2=q^3+q^2+q+1
q^3-3q^2+q+1=0
q^3-2q^2+q-q^2+1=0
q(q-1)^2-(q-1)(q+1)=0
q(q-1)^2-(q-1)(q+1)=0
(q-1)[q(q-1)-(q+1)]=0
(q-1)[q^2-q-q-1]=0
(q-1)[q^2-2q-1]=0
(q-1)[q^2-2q+1-2]=0
(q-1)[(q-1)^2-2]=0
(q-1)[(q-1-√2)(q-1+√2)]=0
(q-1)[(q-1-√2)(q-1+√2)]=0
q=1或q=1+√2或q=1-√2