Question

已知等比數列{a_n}中，若a₁₀₀₅•a₁₀₀₇=4，則該數列的前2011項的積為（　�。�

A．4²⁰¹¹

B．±4²⁰¹¹

C．2²⁰¹¹

D．±2²⁰¹¹

Answer 1

分析：根據等比數列的性質a₁₀₀₅•a₁₀₀₇=a₁•a₂₀₁₁=a₂•a₂₀₁₀=…=a²₁₀₀₆，求出a₁₀₀₆=±2，a₁•a₂•a₃…a₂₀₀₀•a₂₀₀₁=4^1005_•a₁₀₀₆，即可求出結果．

解答：解：∵a₁₀₀₅•a₁₀₀₇=a²₁₀₀₆=4
∴a₁₀₀₆=±2
∴a₁•a₂•a₃…a₂₀₁₀•a₂₀₁₁=（a₁•a₂₀₁₁）•（a₂•a₂₀₁₀）…（a₁₀₀₅•a₁₀₀₇）•a₁₀₀₆=4^1005_•a₁₀₀₆
當a₁₀₀₆=2時a₁•a₂•a₃…a₂₀₁₀•a₂₀₁₁=（a₁•a₂₀₁₁）•（a₂•a₂₀₁₀）…（a₁₀₀₅•a₁₀₀₇）•a₁₀₀₆=4^1005_•a₁₀₀₆=2²⁰¹¹
當a₁₀₀₆=-2時a₁•a₂•a₃…a₂₀₁₀•a₂₀₁₁=（a₁•a₂₀₁₁）•（a₂•a₂₀₁₀）…（a₁₀₀₅•a₁₀₀₇）•a₁₀₀₆=4^1005_•a₁₀₀₆=-2²⁰¹¹
故選D．

點評：本題考查了等比數列的性質，但注意要分類討論，屬于基礎題．