Question

18．設(shè){a_n} 為公比q＞1的等比數(shù)列，若a₂₀₁₃和a₂₀₁₄是方程4x²-8x+3=0的兩根，則a₂₀₁₅+a₂₀₁₆=18．

Answer 1

分析 由4x²-8x+3=0，解得x=$\frac{1}{2}$，$\frac{3}{2}$．根據(jù){a_n} 為公比q＞1的等比數(shù)列，若a₂₀₁₃和a₂₀₁₄是方程4x²-8x+3=0的兩根，可得a₂₀₁₃=$\frac{1}{2}$，a₂₀₁₄=$\frac{3}{2}$．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，若a₂₀₁₃和a₂₀₁₄是方程4x²-8x+3=0的兩根，
∴a₂₀₁₃=$\frac{1}{2}$，a₂₀₁₄=$\frac{3}{2}$，∴q=3．
∴a₂₀₁₅+a₂₀₁₆=q²（a₂₀₁₃+a₂₀₁₄）=18．
故答案為：18．

點(diǎn)評 本題考查了等比數(shù)列的通項(xiàng)公式及其性質(zhì)、一元二次方程的解法，考查了推理能力與計(jì)算能力，屬于中檔題．