Question

已知等比數(shù)列{a_n}中，若a₁₀₀₅•a₁₀₀₇=4，則該數(shù)列的前2011項(xiàng)的積為（　�。�

A．4²⁰¹¹

B．±4²⁰¹¹

C．2²⁰¹¹

D．±2²⁰¹¹

Answer 1

分析：根據(jù)等比數(shù)列的性質(zhì)a₁₀₀₅•a₁₀₀₇=a₁•a₂₀₁₁=a₂•a₂₀₁₀=…=a²₁₀₀₆，求出a₁₀₀₆=±2，a₁•a₂•a₃…a₂₀₀₀•a₂₀₀₁=4^1005_•a₁₀₀₆，即可求出結(jié)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(dāng)a₁₀₀₆=2時(shí)a₁•a₂•a₃…a₂₀₁₀•a₂₀₁₁=（a₁•a₂₀₁₁）•（a₂•a₂₀₁₀）…（a₁₀₀₅•a₁₀₀₇）•a₁₀₀₆=4^1005_•a₁₀₀₆=2²⁰¹¹
當(dāng)a₁₀₀₆=-2時(shí)a₁•a₂•a₃…a₂₀₁₀•a₂₀₁₁=（a₁•a₂₀₁₁）•（a₂•a₂₀₁₀）…（a₁₀₀₅•a₁₀₀₇）•a₁₀₀₆=4^1005_•a₁₀₀₆=-2²⁰¹¹
故選D．

點(diǎn)評(píng)：本題考查了等比數(shù)列的性質(zhì)，但注意要分類討論，屬于基礎(chǔ)題．