若一个等比数列前三项的和为34,最后三项和为146,且所有的项的和为390,则这个数列有几项?

Question

若一个等比数列前三项的和为34,最后三项和为146,且所有的项的和为390,则这个数列有几项?

Answer 1

设an=aq^(n-1)
sn=a1*[q^n-1]/(q-1)=390……(1)
s3=a1*(q^3-1)/(q-1)=34……(2)
a(n-2)+a(n-1)+an=aq^(n-3)+aq^(n-2)+aq^(n-1)
=aq^(n-3)[1+q+q^2]
=q^(n-3)*s3
=34q^(n-3)
=146
q^(n-3)=146/34=73/17……(3)
s(n-3)=sn-[a(n-2)+a(n-1)+an]
=390-146
=244
s(n-3)=a1*[q^(n-3)-1]/(q-1)
=a1*[73/17-1]/(q-1)
=a1*(56/17)/(q-1)
=244
a1/(q-1)=244(17/56)=1037/14
s3=a1*(q^3-1)/(q-1)
=(1037/14)(q^3-1)
=34
q^3=34*14/1037+1=1513/1037
q=(1513/1037)^(1/3)
sn=a1*[q^n-1]/(q-1)
=(1037/14)[q^n-1]
=390
q^n=1+390*14/1037=6497/1037
(1513/1037)^(n/3)=6497/1037
好象没有正整数解.也许算错了,自己算算吧