Question

18．設(shè){a_n}為公比q＞1的等比數(shù)列，若a₂₀₁₄和a₂₀₁₅是方程4x²-8x+3=0的兩根，則a₂₀₁₆+a₂₀₁₇=18．

Answer 1

分析 由4x²-8x+3=0，解得x=$\frac{1}{2}$，$\frac{3}{2}$．由{a_n}為公比q＞1的等比數(shù)列，a₂₀₁₄和a₂₀₁₅是方程4x²-8x+3=0的兩根，可得a₂₀₁₄+a₂₀₁₅=2，a₂₀₁₄=$\frac{1}{2}$，a₂₀₁₅=$\frac{3}{2}$，q=3．即可得出．

解答 解：由4x²-8x+3=0，解得x=$\frac{1}{2}$，$\frac{3}{2}$．
∵{a_n}為公比q＞1的等比數(shù)列，a₂₀₁₄和a₂₀₁₅是方程4x²-8x+3=0的兩根，
∴a₂₀₁₄+a₂₀₁₅=2，a₂₀₁₄=$\frac{1}{2}$，a₂₀₁₅=$\frac{3}{2}$，∴q=3．
則a₂₀₁₆+a₂₀₁₇=q²（a₂₀₁₄+a₂₀₁₅）=3²×2=18．
故答案為：18．

點(diǎn)評(píng) 本題考查了等比數(shù)列的通項(xiàng)公式及其性質(zhì)、數(shù)列遞推關(guān)系、一元二次方程的根與系數(shù)的關(guān)系，考查了推理能力與計(jì)算能力，屬于中檔題．