题目

已知数列an是公比q=1/2的等比数列 且a1+a4+a7+...a100=60 求a1+a2+a3+...a102

提问时间：2020-12-29

答案

a4=a1q^3 a7=a4q^3…… a100=a97q^3
a1,a4,a7,……,a100是以a1为首项,q^3为公比的等比数列,共34项.
S=a1[1-(q^3)^34]/(1-q^3)
=a1(1-q^102)/(1-q^3)
=60
a1(1-q^102)=60(1-q^3)
a1+a2+...+a102
=a1(1-q^102)/(1-q)
=60(1-q^3)/(1-q)
=60(1+q+q^2)
=60[1+(1/2)+(1/2)^2]
=105