Question

一已知一个等比数列前三项积为3,后三项积为9,所有项积为243,问有几项?

Answer 1

a(n)=aq^(n-1),n=1,2,...,N.

3=a*aq*aq^2=(aq)^3,

9=aq^(N-1)*aq^(N-2)*aq^(N-3)=[aq^(N-2)]^3, N>3.

3=9/3=[q^(N-3)]^3=q^(3N-9), q=3^[1/(3N-9)].

q不为1.

3=a^3q^3=a^3*3^[1/(N-3)], a^3=3^[(N-4)/(N-3)], a=3^{(N-4)/[3(N-3)]}

243=3^5=a^N*q^[(N-1)N/2]=3^{(N-4)N/[3(N-3)]}*3^[(N-1)N/(6N-18)]

5=[2N(N-4)+N(N-1)]/(6N-18)=N[3N-9]/(6N-18),

N=10