遞推測試題及答案

單項選擇題（每題2分，共10題）1.數(shù)列1,3,5,7,…的遞推公式可能是（）A.\(a_{n}=a_{n-1}+2\)（\(n\geq2\)）B.\(a_{n}=2a_{n-1}\)（\(n\geq2\)）C.\(a_{n}=a_{n-1}+1\)（\(n\geq2\)）D.\(a_{n}=a_{n-1}-2\)（\(n\geq2\)）2.已知\(a_{1}=1\)，\(a_{n}=a_{n-1}+3\)（\(n\geq2\)），則\(a_{3}\)的值為（）A.5B.7C.9D.113.由\(a_{1}=2\)，\(a_{n+1}=2a_{n}\)，可得\(a_{4}\)等于（）A.8B.16C.32D.644.數(shù)列\(zhòng)(\{a_{n}\}\)滿足\(a_{1}=1\)，\(a_{n+1}=a_{n}+n\)，則\(a_{2}\)為（）A.1B.2C.3D.45.若\(a_{1}=1\)，\(a_{n+1}=3a_{n}\)，則數(shù)列\(zhòng)(\{a_{n}\}\)是（）A.等差數(shù)列B.等比數(shù)列C.既非等差也非等比數(shù)列D.常數(shù)列6.已知\(a_{1}=2\)，\(a_{n}=a_{n-1}+2\)（\(n\geq2\)），則\(a_{5}\)的值為（）A.6B.8C.10D.127.數(shù)列\(zhòng)(\{a_{n}\}\)中，\(a_{1}=3\)，\(a_{n+1}=a_{n}-1\)，則\(a_{4}\)是（）A.0B.1C.2D.38.由\(a_{1}=1\)，\(a_{n+1}=a_{n}+2\)，\(n\inN^\)，則\(a_{10}\)的值為（）A.17B.18C.19D.209.已知數(shù)列\(zhòng)(\{a_{n}\}\)滿足\(a_{1}=4\)，\(a_{n+1}=\frac{1}{2}a_{n}\)，則\(a_{3}\)為（）A.1B.2C.4D.810.數(shù)列\(zhòng)(\{a_{n}\}\)中\(zhòng)(a_{1}=1\)，\(a_{n+1}=a_{n}+3\)，該數(shù)列的通項公式\(a_{n}\)是（）A.\(3n-2\)B.\(3n+2\)C.\(2n-3\)D.\(2n+3\)多項選擇題（每題2分，共10題）1.下列關(guān)于遞推數(shù)列的說法正確的是（）A.遞推數(shù)列一定有通項公式B.由遞推公式可以確定數(shù)列的每一項C.遞推公式包含初始值和遞推關(guān)系D.有些遞推數(shù)列是周期數(shù)列2.已知數(shù)列\(zhòng)(\{a_{n}\}\)滿足\(a_{1}=1\)，\(a_{n+1}=a_{n}+2\)，則（）A.數(shù)列\(zhòng)(\{a_{n}\}\)是等差數(shù)列B.\(a_{n}=2n-1\)C.\(a_{5}=9\)D.該數(shù)列的公差為23.數(shù)列\(zhòng)(\{a_{n}\}\)中，\(a_{1}=2\)，\(a_{n+1}=2a_{n}\)，則（）A.數(shù)列\(zhòng)(\{a_{n}\}\)是等比數(shù)列B.\(a_{n}=2^n\)C.\(a_{4}=16\)D.公比為24.對于數(shù)列\(zhòng)(\{a_{n}\}\)，\(a_{1}=3\)，\(a_{n+1}=a_{n}-1\)，以下說法正確的是（）A.數(shù)列\(zhòng)(\{a_{n}\}\)是遞減數(shù)列B.\(a_{n}=4-n\)C.\(a_{3}=1\)D.數(shù)列\(zhòng)(\{a_{n}\}\)是等差數(shù)列5.已知數(shù)列\(zhòng)(\{a_{n}\}\)的遞推公式\(a_{1}=1\)，\(a_{n+1}=a_{n}+n\)，則（）A.\(a_{2}=2\)B.\(a_{3}=4\)C.\(a_{4}=7\)D.\(a_{5}=11\)6.數(shù)列\(zhòng)(\{a_{n}\}\)滿足\(a_{1}=5\)，\(a_{n+1}=3a_{n}\)，則（）A.數(shù)列\(zhòng)(\{a_{n}\}\)是等比數(shù)列B.\(a_{n}=5\times3^{n-1}\)C.\(a_{3}=45\)D.公比是37.若數(shù)列\(zhòng)(\{a_{n}\}\)的遞推關(guān)系為\(a_{1}=1\)，\(a_{n+1}=2a_{n}+1\)，則（）A.\(a_{2}=3\)B.\(a_{3}=7\)C.可通過構(gòu)造新數(shù)列求\(a_{n}\)D.\(a_{n}=2^n-1\)8.下列遞推公式能確定數(shù)列的有（）A.\(a_{1}=1\)，\(a_{n+1}=a_{n}+1\)B.\(a_{1}=2\)，\(a_{n+1}=2a_{n}\)C.\(a_{1}=3\)，\(a_{n+1}=a_{n}^2\)D.\(a_{1}=4\)，\(a_{n+1}=a_{n}+n\)9.數(shù)列\(zhòng)(\{a_{n}\}\)中\(zhòng)(a_{1}=-1\)，\(a_{n+1}=a_{n}-2\)，則（）A.數(shù)列\(zhòng)(\{a_{n}\}\)是等差數(shù)列B.\(a_{n}=-2n+1\)C.\(a_{3}=-5\)D.公差為-210.已知數(shù)列\(zhòng)(\{a_{n}\}\)滿足\(a_{1}=1\)，\(a_{n+1}=a_{n}+\frac{1}{n(n+1)}\)，則（）A.\(a_{2}=\frac{3}{2}\)B.\(a_{3}=\frac{5}{3}\)C.\(a_{n}=2-\frac{1}{n}\)D.該數(shù)列是遞增數(shù)列判斷題（每題2分，共10題）1.只要有遞推公式就能唯一確定一個數(shù)列。（）2.若\(a_{1}=1\)，\(a_{n+1}=a_{n}+1\)，則\(a_{n}=n\)。（）3.遞推數(shù)列一定是無窮數(shù)列。（）4.數(shù)列\(zhòng)(\{a_{n}\}\)中\(zhòng)(a_{1}=2\)，\(a_{n+1}=2a_{n}\)，則\(a_{n}=2^n\)。（）5.由\(a_{1}=3\)，\(a_{n+1}=a_{n}-1\)可知數(shù)列\(zhòng)(\{a_{n}\}\)是遞增數(shù)列。（）6.若數(shù)列遞推公式中沒有初始值，則無法確定數(shù)列。（）7.已知\(a_{1}=1\)，\(a_{n+1}=a_{n}+n\)，\(a_{3}=4\)。（）8.遞推公式只能用\(a_{n+1}\)與\(a_{n}\)的關(guān)系表示。（）9.數(shù)列\(zhòng)(\{a_{n}\}\)中\(zhòng)(a_{1}=4\)，\(a_{n+1}=\frac{1}{2}a_{n}\)，該數(shù)列是等比數(shù)列。（）10.若\(a_{1}=-1\)，\(a_{n+1}=a_{n}-2\)，則\(a_{n}=-2n+1\)。（）簡答題（每題5分，共4題）1.已知數(shù)列\(zhòng)(\{a_{n}\}\)滿足\(a_{1}=1\)，\(a_{n+1}=a_{n}+2\)，求\(a_{n}\)。答案：由\(a_{n+1}-a_{n}=2\)，可知數(shù)列\(zhòng)(\{a_{n}\}\)是首項\(a_{1}=1\)，公差\(d=2\)的等差數(shù)列。根據(jù)等差數(shù)列通項公式\(a_{n}=a_{1}+(n-1)d\)，可得\(a_{n}=1+2(n-1)=2n-1\)。2.數(shù)列\(zhòng)(\{a_{n}\}\)中\(zhòng)(a_{1}=2\)，\(a_{n+1}=3a_{n}\)，求\(a_{n}\)。答案：因為\(\frac{a_{n+1}}{a_{n}}=3\)，所以數(shù)列\(zhòng)(\{a_{n}\}\)是首項\(a_{1}=2\)，公比\(q=3\)的等比數(shù)列。根據(jù)等比數(shù)列通項公式\(a_{n}=a_{1}q^{n-1}\)，可得\(a_{n}=2\times3^{n-1}\)。3.已知\(a_{1}=1\)，\(a_{n+1}=a_{n}+n\)，求\(a_{4}\)。答案：\(a_{2}=a_{1}+1=1+1=2\)；\(a_{3}=a_{2}+2=2+2=4\)；\(a_{4}=a_{3}+3=4+3=7\)。4.數(shù)列\(zhòng)(\{a_{n}\}\)滿足\(a_{1}=3\)，\(a_{n+1}=a_{n}-1\)，判斷數(shù)列類型并求\(a_{n}\)。答案：由于\(a_{n+1}-a_{n}=-1\)，所以數(shù)列\(zhòng)(\{a_{n}\}\)是首項\(a_{1}=3\)，公差\(d=-1\)的等差數(shù)列。\(a_{n}=a_{1}+(n-1)d=3+(n-1)\times(-1)=4-n\)。討論題（每題5分，共4題）1.討論遞推數(shù)列在實際生活中的應用。答案：遞推數(shù)列在很多實際場景有應用，如經(jīng)濟領域的貸款還款計算，通過遞推關(guān)系計算每期還款額和欠款余額；又如生物繁殖問題，根據(jù)種群繁殖規(guī)律建立遞推模型預測種群數(shù)量變化，能輔助制定合理決策。2.當一個數(shù)列有多種遞推表示方法時，如何選擇最優(yōu)的？答案：選擇最優(yōu)遞推表示方法，需看目的。若求通項公式，選易于轉(zhuǎn)化為已知數(shù)列類型（等差、等比）的；若用于計算機編程計算數(shù)列項，選計算復雜度低、迭代步驟簡單的；若分析數(shù)列性質(zhì)，選能清晰體現(xiàn)性質(zhì)的遞推式。3.探討遞推數(shù)列與函數(shù)的關(guān)系。答案：遞推數(shù)列可看作是定義域為正整數(shù)集的離散函數(shù)。數(shù)列的遞推關(guān)系類似于函數(shù)的迭代關(guān)系。通過函數(shù)觀點能分析數(shù)列單調(diào)性、周期性等性質(zhì)，例如借助函數(shù)圖像判斷數(shù)列增減，利用函數(shù)周期性概念理解數(shù)列周期情況。4.