已知数列{An}为等比数列S6=21√3,S2=√3,求S9

设第一项为a1，比值为q
S2=a1*(1-q^2)/(1-q)=√3
S6=a1*(1-q^6)/(1-q)=21√3
S9=a1*(1-q^9)/(1-q)

三式联立，可解得q=2，a1=√3/3，S9=511√3/3

S6=a1(1-q^6)/(1-q)=21√3
S2=a1(1-q^2)/(1-q)=√3
S6/S2=(1-q^6)/(1-q^2)=(1-q^2)(1+q^2+q^4)/(1-q^2)=1+q^2+q^4=21
即q^4+q^2-20=0 即(q^2+5)(q^2-4)=0
看出q^2=4 q=2或q=-2
q=2时 a1(1-2^6)/(1-2)=21√3 解得a1=√3/3
S9=a1(1-q^9)/(1-q)=√3/3(1-2^9)/(1-2)=511√3/3

q=-2时a1(1-2^6)/(1+2)=21√3 解得a1=-√3
S9=a1(1-q^9)/(1-q)=-√3(1+2^9)/(1+2)=-513√3/3

S6=a1q^5

S2=a1q

S6/S2=q^4=21

q=21^(1/4)

S2=a1q=√3

a1=3^(1/2)/21^(1/4)=3^(1/4)/7^(1/4)=(3/7)^(1/4)

S9=a1q^8=(3/7)^(1/4)*21^2=441*(3/7)^(1/4)

{An}为等比数列,则
S4=√(S2*S6)=3√7
S5=√(S4*S6)=3*7^(1/2)*21^(1/4)
S9=S5*S6/S2=63*7^(1/2)*21^(1/4)