專(zhuān)題06構(gòu)造法求數(shù)列通項(xiàng)的八種技巧(三)【必備知識(shí)點(diǎn)】◆構(gòu)造六:取對(duì)數(shù)構(gòu)造法型如SKIPIF1<0,SKIPIF1<0或者SKIPIF1<0為常數(shù).針對(duì)出現(xiàn)這種數(shù)列,為方便計(jì)算,兩邊通常取以SKIPIF1<0或首項(xiàng)為底的對(duì)數(shù),就能找到突破口.什么情況取SKIPIF1<0為底,什么情況取首項(xiàng)為底呢?我們來(lái)看兩道例題.【經(jīng)典例題1】數(shù)列SKIPIF1<0中,SKIPIF1<0,SKIPIF1<0,求數(shù)列SKIPIF1<0的通項(xiàng)公式.【經(jīng)典例題2】數(shù)列SKIPIF1<0中,SKIPIF1<0,SKIPIF1<0,求數(shù)列SKIPIF1<0的通項(xiàng)公式.【經(jīng)典例題3】已知SKIPIF1<0,點(diǎn)SKIPIF1<0在函數(shù)SKIPIF1<0的圖像上,其中SKIPIF1<0,求數(shù)列SKIPIF1<0的通項(xiàng)公式.【經(jīng)典例題4】在數(shù)列SKIPIF1<0中,SKIPIF1<0,當(dāng)SKIPIF1<0時(shí),有SKIPIF1<0,求數(shù)列SKIPIF1<0的通項(xiàng)公式.◆構(gòu)造七:二階整體構(gòu)造等比簡(jiǎn)單的二階整體等比:關(guān)于SKIPIF1<0的模型,可通過(guò)構(gòu)造二階等比數(shù)列求解,大部分題型可轉(zhuǎn)化為SKIPIF1<0,利用SKIPIF1<0成等比數(shù)列,以及疊加法求出SKIPIF1<0.還有一小部分題型可轉(zhuǎn)化為SKIPIF1<0,利用SKIPIF1<0成等比數(shù)列求出SKIPIF1<0.【經(jīng)典例題1】已知數(shù)列SKIPIF1<0滿(mǎn)足SKIPIF1<0,求數(shù)列SKIPIF1<0的通項(xiàng)公式.【經(jīng)典例題2】已知數(shù)列SKIPIF1<0中,SKIPIF1<0,SKIPIF1<0,SKIPIF1<0,求數(shù)列SKIPIF1<0的通項(xiàng)公式。【經(jīng)典例題3】數(shù)列SKIPIF1<0中,SKIPIF1<0,SKIPIF1<0,SKIPIF1<0,求數(shù)列SKIPIF1<0的通項(xiàng)公式。此方法可以解決大多數(shù)的SKIPIF1<0,SKIPIF1<0模型的試題.當(dāng)然針對(duì)個(gè)別試題,單純構(gòu)造SKIPIF1<0成等比數(shù)列可能解決不了問(wèn)題.我們需要學(xué)習(xí)更完整的方法來(lái)解決這種類(lèi)型題.這就需要運(yùn)用數(shù)列的特征方程理念來(lái)解決.當(dāng)然我們不需要詳細(xì)學(xué)習(xí)數(shù)列的特征方程,用高中的待定系數(shù)法也可以解決,接下來(lái)我們通過(guò)兩道例題,來(lái)詳細(xì)解釋說(shuō)明下這種方法.【經(jīng)典例題4】已知數(shù)列SKIPIF1<0滿(mǎn)足SKIPIF1<0,SKIPIF1<0,SKIPIF1<0,求數(shù)列SKIPIF1<0的通項(xiàng)公式.【經(jīng)典例題5】已知數(shù)列SKIPIF1<0滿(mǎn)足SKIPIF1<0,SKIPIF1<0,SKIPIF1<0,求SKIPIF1<0的通項(xiàng)公式.秒殺求法:SKIPIF1<0類(lèi)通項(xiàng)公式暴力秒殺求法SKIPIF1<0對(duì)應(yīng)的特征方程為:SKIPIF1<0,設(shè)其兩根為SKIPIF1<0當(dāng)SKIPIF1<0時(shí),SKIPIF1<0當(dāng)SKIPIF1<0時(shí),SKIPIF1<0其中SKIPIF1<0,SKIPIF1<0的值的求法,用SKIPIF1<0的值代入上面的通項(xiàng)公式中,建立方程組解之即可【秒殺例題1】已知數(shù)列SKIPIF1<0滿(mǎn)足SKIPIF1<0,SKIPIF1<0,SKIPIF1<0,求SKIPIF1<0的通項(xiàng)公式.【秒殺例題2】已知數(shù)列SKIPIF1<0滿(mǎn)足SKIPIF1<0,SKIPIF1<0,SKIPIF1<0,求數(shù)列SKIPIF1<0的通項(xiàng)公式.【練習(xí)1】在數(shù)列SKIPIF1<0中,SKIPIF1<0,則SKIPIF1<0_______.【練習(xí)2】設(shè)數(shù)列SKIPIF1<0的前SKIPIF1<0項(xiàng)和為SKIPIF1<0.已知SKIPIF1<0,且當(dāng)SKIPIF1<0時(shí),SKIPIF1<0.(1)求SKIPIF1<0的值；(2)證明：SKIPIF1<0為等比數(shù)列;(3)求數(shù)列SKIPIF1<0的通項(xiàng)公式.【練習(xí)3】數(shù)列SKIPIF1<0滿(mǎn)足SKIPIF1<0.(1)設(shè)SKIPIF1<0,證明SKIPIF1<0是等差數(shù)列;(2)求SKIPIF1<0的通項(xiàng)公式.◆構(gòu)造八：數(shù)列不動(dòng)點(diǎn)構(gòu)造求數(shù)列(較難,能力強(qiáng)的同學(xué)可以學(xué)習(xí))針對(duì)SKIPIF1<0這類(lèi)題型,考題中并不多見(jiàn),難度比較大,這類(lèi)題型有特定的解題方法.我們需要學(xué)習(xí)“數(shù)列不動(dòng)點(diǎn)”的知識(shí)點(diǎn).接下來(lái)我們來(lái)學(xué)習(xí)下什么是“數(shù)列不動(dòng)點(diǎn)”,它有什么性質(zhì).當(dāng)然看不懂也沒(méi)關(guān)系,可以通過(guò)例題,熟記掌握解題步驟就可以.對(duì)于函數(shù)SKIPIF1<0,若存在實(shí)數(shù)SKIPIF1<0,使得SKIPIF1<0,則稱(chēng)SKIPIF1<0是函數(shù)SKIPIF1<0的不動(dòng)點(diǎn).在幾何上，曲線SKIPIF1<0與曲線SKIPIF1<0的交點(diǎn)的橫坐標(biāo)即為函數(shù)SKIPIF1<0的不動(dòng)點(diǎn).一般地,數(shù)列SKIPIF1<0的遞推式可以由公式SKIPIF1<0給出,因此可以定義遞推數(shù)列的不動(dòng)點(diǎn):對(duì)于遞推數(shù)列SKIPIF1<0,若其遞推式為SKIPIF1<0,且存在實(shí)數(shù)SKIPIF1<0,使得SKIPIF1<0,則稱(chēng)SKIPIF1<0是數(shù)列SKIPIF1<0的不動(dòng)點(diǎn)。數(shù)列的不動(dòng)點(diǎn)有什么性質(zhì)呢?若從某一項(xiàng)SKIPIF1<0開(kāi)始,數(shù)列的取值即為SKIPIF1<0,也即SKIPIF1<0,則SKIPIF1<0,以此類(lèi)推,根據(jù)數(shù)學(xué)歸納法,可以得到當(dāng)SKIPIF1<0時(shí),SKIPIF1<0,也即數(shù)列SKIPIF1<0在SKIPIF1<0之后“不動(dòng)”了.這就為我們求數(shù)列的不動(dòng)點(diǎn)提供一個(gè)思路,當(dāng)數(shù)列達(dá)到不動(dòng)點(diǎn),之后的每一項(xiàng)都相等,所以在給定等式中,令數(shù)列當(dāng)中的每一項(xiàng)都等于SKIPIF1<0,最后解方程即可.接下來(lái)我們來(lái)看分式遞推數(shù)列,這也是不動(dòng)點(diǎn)法主要應(yīng)用的范圍.所謂分式遞推數(shù)列是指以下類(lèi)型：若數(shù)列SKIPIF1<0滿(mǎn)足SKIPIF1<0,其中SKIPIF1<0,SKIPIF1<0,SKIPIF1<0,SKIPIF1<0是給定的實(shí)數(shù),求數(shù)列SKIPIF1<0的通項(xiàng)公式。這時(shí)候要求它的不動(dòng)點(diǎn),考慮方程SKIPIF1<0,得到了一個(gè)二次方程,我們從幾個(gè)例子出發(fā)：【經(jīng)典例題1】設(shè)數(shù)列SKIPIF1<0滿(mǎn)足SKIPIF1<0,求數(shù)列SKIPIF1<0的通項(xiàng)公式。【經(jīng)典例題2】設(shè)數(shù)列SKIPIF1<0滿(mǎn)足SKIPIF1<0,求數(shù)列SKIPIF1<0的通項(xiàng)公式?！窘?jīng)典例題3】已知SKIPIF1<0,且SKIPIF1<0,求SKIPIF1<0的通項(xiàng)公式.【經(jīng)典例題4】數(shù)列SKIPIF1<0滿(mǎn)足SKIPIF1<0,求SKIPIF1<0的通項(xiàng)公式【經(jīng)典例題5】設(shè)數(shù)列SKIPIF1<0滿(mǎn)足SKIPIF1<0,求數(shù)列SKIPIF1<0的通項(xiàng)公式?？偨Y(jié):形如SKIPIF1<0的遞推數(shù)列,首先令SKIPIF1<0,解出數(shù)列的不動(dòng)點(diǎn).處理時(shí)也可以分兩種情況:(1)若其有一個(gè)不動(dòng)點(diǎn)SKIPIF1<0,則SKIPIF1<0是等差數(shù)列;(2)若其有兩個(gè)不動(dòng)點(diǎn)SKIPIF1<0,則SKIPIF1<0是等比數(shù)列?！具^(guò)關(guān)檢測(cè)】一、單選題1．已知數(shù)列SKIPIF1<0的前SKIPIF1<0項(xiàng)和為SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，則SKIPIF1<0（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<02．在數(shù)列SKIPIF1<0中，SKIPIF1<0，SKIPIF1<0，則SKIPIF1<0的值為(

)A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．無(wú)法確定3．已知數(shù)列SKIPIF1<0滿(mǎn)足：SKIPIF1<0，SKIPIF1<0，則SKIPIF1<0（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<04．已知數(shù)列SKIPIF1<0滿(mǎn)足SKIPIF1<0，且SKIPIF1<0，SKIPIF1<0，則SKIPIF1<0（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<05．已知數(shù)列SKIPIF1<0滿(mǎn)足SKIPIF1<0，且SKIPIF1<0，SKIPIF1<0，其前n項(xiàng)和為SKIPIF1<0，若對(duì)任意的正整數(shù)n，SKIPIF1<0恒成立，則m的取值范圍是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<06．已知數(shù)列SKIPIF1<0，SKIPIF1<0，則SKIPIF1<0（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<07．已知數(shù)列SKIPIF1<0滿(mǎn)足SKIPIF1<0，SKIPIF1<0，且SKIPIF1<0，若SKIPIF1<0表示不超過(guò)x的最大整數(shù)（例如SKIPIF1<0，SKIPIF1<0）.則SKIPIF1<0（

）A．2018 B．2019 C．2020 D．2021二、填空題8．在數(shù)列SKIPIF1<0中，SKIPIF1<0，SKIPIF1<0，且滿(mǎn)足SKIPIF1<0SKIPIF1<0，則SKIPIF1<0___________.9．已知數(shù)列SKIPIF1<0滿(mǎn)足SKIPIF1<0，且SKIPIF1<0，則SKIPIF1<0的通項(xiàng)公式SKIPIF1<0_______________________.10．設(shè)正項(xiàng)數(shù)列SKIPIF1<0