數(shù)學(xué)數(shù)列高考題目及答案

單項選擇題（每題2分，共10題）1.等差數(shù)列\(zhòng)(\{a_{n}\}\)中，\(a_{1}=2\)，\(a_{3}=6\)，則公差\(d\)為（）A.1B.2C.3D.42.等比數(shù)列\(zhòng)(\{a_{n}\}\)中，\(a_{1}=1\)，\(a_{2}=2\)，則公比\(q\)是（）A.1B.2C.3D.43.數(shù)列\(zhòng)(\{a_{n}\}\)通項公式\(a_{n}=3n-1\)，則\(a_{5}\)的值為（）A.14B.15C.16D.174.等差數(shù)列\(zhòng)(\{a_{n}\}\)前\(n\)項和\(S_{n}\)，若\(a_{1}=1\)，\(S_{3}=9\)，則\(a_{3}\)為（）A.3B.4C.5D.65.等比數(shù)列\(zhòng)(\{a_{n}\}\)中，\(a_{3}=4\)，\(a_{5}=16\)，則\(a_{4}\)等于（）A.8B.-8C.\(\pm8\)D.106.數(shù)列\(zhòng)(\{a_{n}\}\)滿足\(a_{n+1}=a_{n}+2\)，\(a_{1}=1\)，則\(a_{4}\)為（）A.5B.7C.9D.117.等差數(shù)列\(zhòng)(\{a_{n}\}\)中，\(a_{2}+a_{4}=10\)，則\(a_{3}\)的值是（）A.5B.6C.7D.88.等比數(shù)列\(zhòng)(\{a_{n}\}\)中，\(a_{1}a_{9}=25\)，則\(a_{5}\)的值為（）A.5B.-5C.\(\pm5\)D.109.數(shù)列\(zhòng)(\{a_{n}\}\)通項\(a_{n}=2^{n}\)，則\(a_{3}\)是（）A.4B.6C.8D.1010.等差數(shù)列\(zhòng)(\{a_{n}\}\)中，\(S_{n}\)為前\(n\)項和，若\(a_{1}=2\)，\(S_{2}=6\)，則\(S_{3}\)為（）A.10B.12C.14D.16多項選擇題（每題2分，共10題）1.以下哪些是等差數(shù)列的性質(zhì)（）A.\(a_{n}=a_{1}+(n-1)d\)B.\(a_{m}+a_{n}=a_{p}+a_{q}(m+n=p+q)\)C.\(S_{n}=\frac{n(a_{1}+a_{n})}{2}\)D.\(a_{n}=a_{1}q^{n-1}\)2.等比數(shù)列\(zhòng)(\{a_{n}\}\)中，可能成立的是（）A.\(a_{1}=1\)，\(q=2\)B.\(a_{1}=-1\)，\(q=-1\)C.\(a_{1}=0\)，\(q=2\)D.\(a_{1}=1\)，\(q=0\)3.數(shù)列\(zhòng)(\{a_{n}\}\)通項公式\(a_{n}=n^{2}\)，則（）A.\(a_{1}=1\)B.\(a_{2}=4\)C.\(a_{3}=9\)D.\(a_{4}=16\)4.等差數(shù)列\(zhòng)(\{a_{n}\}\)中，若\(a_{3}=5\)，\(d=2\)，則（）A.\(a_{1}=1\)B.\(a_{2}=3\)C.\(a_{4}=7\)D.\(a_{5}=9\)5.等比數(shù)列\(zhòng)(\{a_{n}\}\)中，\(a_{1}=1\)，\(a_{3}=4\)，則公比\(q\)的值可能是（）A.2B.-2C.4D.-46.下列數(shù)列中，哪些是遞增數(shù)列（）A.\(a_{n}=n\)B.\(a_{n}=-n\)C.\(a_{n}=2^{n}\)D.\(a_{n}=(\frac{1}{2})^{n}\)7.等差數(shù)列\(zhòng)(\{a_{n}\}\)的前\(n\)項和\(S_{n}=n^{2}\)，則（）A.\(a_{1}=1\)B.\(a_{2}=3\)C.\(a_{3}=5\)D.\(a_{4}=7\)8.等比數(shù)列\(zhòng)(\{a_{n}\}\)滿足\(a_{2}a_{8}=16\)，則（）A.\(a_{5}^{2}=16\)B.\(a_{5}=4\)C.\(a_{5}=-4\)D.\(a_{5}=\pm4\)9.數(shù)列\(zhòng)(\{a_{n}\}\)中，\(a_{n+1}-a_{n}=3\)，\(a_{1}=1\)，則（）A.數(shù)列\(zhòng)(\{a_{n}\}\)是等差數(shù)列B.\(a_{2}=4\)C.\(a_{3}=7\)D.\(a_{4}=10\)10.已知數(shù)列\(zhòng)(\{a_{n}\}\)是等比數(shù)列，公比\(q\gt1\)，則（）A.數(shù)列\(zhòng)(\{a_{n}\}\)可能是遞增數(shù)列B.數(shù)列\(zhòng)(\{a_{n}\}\)可能是遞減數(shù)列C.\(a_{n+1}\gta_{n}\)D.\(a_{n+1}\lta_{n}\)判斷題（每題2分，共10題）1.常數(shù)列一定是等差數(shù)列。（）2.常數(shù)列一定是等比數(shù)列。（）3.若\(a,b,c\)成等差數(shù)列，則\(2b=a+c\)。（）4.若\(a,b,c\)成等比數(shù)列，則\(b^{2}=ac\)。（）5.等差數(shù)列\(zhòng)(\{a_{n}\}\)的前\(n\)項和\(S_{n}\)一定是關(guān)于\(n\)的二次函數(shù)。（）6.等比數(shù)列\(zhòng)(\{a_{n}\}\)中，\(a_{n}=a_{m}q^{n-m}\)。（）7.數(shù)列\(zhòng)(\{a_{n}\}\)通項\(a_{n}=(-1)^{n}\)是擺動數(shù)列。（）8.等差數(shù)列\(zhòng)(\{a_{n}\}\)中，若\(a_{1}\gt0\)，\(d\lt0\)，則數(shù)列遞減。（）9.等比數(shù)列\(zhòng)(\{a_{n}\}\)中，若\(a_{1}\gt0\)，\(0\ltq\lt1\)，則數(shù)列遞減。（）10.數(shù)列\(zhòng)(\{a_{n}\}\)滿足\(a_{n+1}=a_{n}\)，則該數(shù)列是常數(shù)列。（）簡答題（每題5分，共4題）1.已知等差數(shù)列\(zhòng)(\{a_{n}\}\)中，\(a_{1}=3\)，\(d=2\)，求\(a_{n}\)。答案：根據(jù)等差數(shù)列通項公式\(a_{n}=a_{1}+(n-1)d\)，將\(a_{1}=3\)，\(d=2\)代入得\(a_{n}=3+2(n-1)=2n+1\)。2.等比數(shù)列\(zhòng)(\{a_{n}\}\)中，\(a_{1}=2\)，\(q=3\)，求\(a_{4}\)。答案：由等比數(shù)列通項公式\(a_{n}=a_{1}q^{n-1}\)，可得\(a_{4}=a_{1}q^{3}=2×3^{3}=2×27=54\)。3.已知等差數(shù)列\(zhòng)(\{a_{n}\}\)前\(n\)項和\(S_{n}=n^{2}+n\)，求\(a_{1}\)和\(a_{2}\)。答案：\(n=1\)時，\(a_{1}=S_{1}=1^{2}+1=2\)；\(n=2\)時，\(S_{2}=2^{2}+2=6\)，\(a_{2}=S_{2}-S_{1}=6-2=4\)。4.等比數(shù)列\(zhòng)(\{a_{n}\}\)中，\(a_{2}=4\)，\(a_{4}=16\)，求公比\(q\)。答案：由等比數(shù)列性質(zhì)\(a_{4}=a_{2}q^{2}\)，將\(a_{2}=4\)，\(a_{4}=16\)代入得\(16=4q^{2}\)，\(q^{2}=4\)，解得\(q=\pm2\)。討論題（每題5分，共4題）1.討論等差數(shù)列與一次函數(shù)的關(guān)系。答案：等差數(shù)列\(zhòng)(a_{n}=a_{1}+(n-1)d=dn+(a_{1}-d)\)，其通項公式是關(guān)于\(n\)的一次函數(shù)形式（\(d\neq0\)時），\(d\)是斜率；\(d=0\)時為常數(shù)列。前\(n\)項和\(S_{n}=An^{2}+Bn\)（\(A=\fracmog2w40{2}\)，\(B=a_{1}-\fraciqsmcy4{2}\)）是關(guān)于\(n\)的二次函數(shù)（\(d\neq0\)）。2.等比數(shù)列的單調(diào)性受哪些因素影響？答案：當(dāng)\(a_{1}\gt0\)，\(q\gt1\)或\(a_{1}\lt0\)，\(0\ltq\lt1\)時，等比數(shù)列遞增；當(dāng)\(a_{1}\gt0\)，\(0\ltq\lt1\)或\(a_{1}\lt0\)，\(q\gt1\)時，等比數(shù)列遞減；當(dāng)\(q=1\)時為常數(shù)列；當(dāng)\(q\lt0\)時為擺動數(shù)列。3.如何判斷一個數(shù)列是等差數(shù)列還是等比數(shù)列？答案：判斷等差數(shù)列可看\(a_{n+1}-a_{n}\)是否為常數(shù)，或\(2a_{n+1}=a_{n}+a_{n+2}\)是否成立。判斷等比數(shù)列可看\(\frac{a_{n+1}}{a_{n}}\)是否為非零常數(shù)，或\(a_{n+1}^{2}=a_{n}a_{n+2}\neq0\)是否成立。4.數(shù)列在實際生活中有哪些應(yīng)用？答案：數(shù)列在貸款、儲蓄、人口增長、資源利用等方面有應(yīng)用。如貸款還款計劃可用等差數(shù)列計算每期還款額；儲蓄復(fù)利計算、細胞分裂等符合等比數(shù)列模型，通過數(shù)列知識能進行預(yù)測和分析。答案單項選擇