正数组成的等比数列{an}公比q=2,a1×a2×a3...×a30=2^30,求a3×a6×a9...×a30=A1B2^10C2^15D2^20

Question

正数组成的等比数列{an}公比q=2,a1×a2×a3...×a30=2^30,求a3×a6×a9...×a30= A1B2^10C2^15D2^20

Answer 1

答案D2^20
a1
a2=a1×q
a3；a4=a3×q；a5=a3×q^2；所以a3×a4×a5＝a3^3×q^3
同理a6；a7=a6×q；a8=a6×q^2；所以a6×a7×a8＝a6^3×q^3
a9；a10=a9×q；a11=a9×q^2；所以a9×a10×a11＝a9^3×q^3
.
a27；a28=a27×q；a29=a27×q^2；所以a27×a28×a29＝a27^3×q^3
a30=a30^3×q^3/a30^2×q^3=a30^3×q^3/a1^2×q^61
a1×a2×a3...×a30
＝a1^2×q×(a3^3×q^3)×(a6^3×q^3)×(a9^3×q^3)×.×(a27^3×q^3)×a30^3×q^3/a1^2×q^61
=a1^2×q×(a3×a6×a9...×a30)^3×q^30/a1^2×q^61
因为a1×a2×a3...×a30=2^30 且q=2
所以（a3×a6×a9...×a30)^3＝2^90/2^30=2^60=(2^20)^3
a3×a6×a9...×a30=2^20所以应先D