Question

1.在等比数列an中,a5*a6*a7=3,a6*a7*a8=24,a7*a8*a9等于多少?

Answer 1

1.a6*a7*a8=a5*q*a6*q*a7*q=a5*a6*a7*q³=3q³=24

所以公比q=2

a7*a8*a9=a6*q*a7*q*a8*q=24*2³=192

2.S5=10=a1*(1-q^5)/(1-q)

S10=50=a1*(1-q^10)/(1-q)

所以10/50=(1-q^5)/(1-q^10)

则(q^5)²-5*q^5+4=0

(q^5-1)(q^5-4)=0

解得q^5=1或q^5=4

当q^5=1时,S5=5a1=10,a1=2,S10=10a1=20≠50,故q≠1

当q^5=4时,S5=a1*(1-q^5)/(1-q)=a1*(-3)/(1-q)=10,所以a1/(1-q)=-10/3

S15=a1*(1-q^15)/(1-q)=a1/(1-q)*(1-q^15)=-10/3*[1-(q^5)³]=-10/3*(1-q^5)(1+q^5+q^10)

=-10/3*(1-4)*(1+4+4²)=210

即S15=210