Question

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

Answer 1

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