新高考數(shù)學(xué)一輪復(fù)習(xí)百題刷過(guò)關(guān)專(zhuān)題16 數(shù)列放縮證明不等式必刷100題(原卷版)_第1頁(yè)
新高考數(shù)學(xué)一輪復(fù)習(xí)百題刷過(guò)關(guān)專(zhuān)題16 數(shù)列放縮證明不等式必刷100題(原卷版)_第2頁(yè)
新高考數(shù)學(xué)一輪復(fù)習(xí)百題刷過(guò)關(guān)專(zhuān)題16 數(shù)列放縮證明不等式必刷100題(原卷版)_第3頁(yè)
新高考數(shù)學(xué)一輪復(fù)習(xí)百題刷過(guò)關(guān)專(zhuān)題16 數(shù)列放縮證明不等式必刷100題(原卷版)_第4頁(yè)
新高考數(shù)學(xué)一輪復(fù)習(xí)百題刷過(guò)關(guān)專(zhuān)題16 數(shù)列放縮證明不等式必刷100題(原卷版)_第5頁(yè)
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專(zhuān)題16數(shù)列放縮證明不等式必刷100題任務(wù)一:邪惡模式(困難)1-100題提示:幾種常見(jiàn)的數(shù)列放縮方法:(1)SKIPIF1<0;(2)SKIPIF1<0;(3)SKIPIF1<0;(4)SKIPIF1<0;(5)SKIPIF1<0;(6)SKIPIF1<0;(7)SKIPIF1<0;(8)SKIPIF1<0SKIPIF1<0;(9)SKIPIF1<0SKIPIF1<0SKIPIF1<0;(10)SKIPIF1<0SKIPIF1<0;(11)SKIPIF1<0;(12)SKIPIF1<0.一、單選題1.2018年9月24日,英國(guó)數(shù)學(xué)家M.F阿帝亞爵在“海德堡論壇”展示了他“證明”黎曼猜想的過(guò)程,引起數(shù)學(xué)界震動(dòng),黎曼猜想來(lái)源于一些特殊數(shù)列求和.記無(wú)窮數(shù)列SKIPIF1<0的各項(xiàng)的和SKIPIF1<0,那么下列結(jié)論正確的是A.SKIPIF1<0 B.SKIPIF1<0 C.SKIPIF1<0 D.SKIPIF1<02.已知數(shù)列SKIPIF1<0滿(mǎn)足SKIPIF1<0,SKIPIF1<0,且SKIPIF1<0,SKIPIF1<0,則下列說(shuō)法中錯(cuò)誤的是()A.SKIPIF1<0 B.SKIPIF1<0C.SKIPIF1<0 D.SKIPIF1<03.已知數(shù)列SKIPIF1<0滿(mǎn)足SKIPIF1<0,SKIPIF1<0,則下列選項(xiàng)正確的是()A.SKIPIF1<0 B.SKIPIF1<0C.SKIPIF1<0 D.SKIPIF1<04.已知數(shù)列SKIPIF1<0滿(mǎn)足SKIPIF1<0,SKIPIF1<0,若SKIPIF1<0,對(duì)任意的SKIPIF1<0,SKIPIF1<0恒成立,則SKIPIF1<0的最小值為().A.SKIPIF1<0 B.SKIPIF1<0 C.SKIPIF1<0 D.35.已知數(shù)列SKIPIF1<0的前SKIPIF1<0項(xiàng)和為SKIPIF1<0,滿(mǎn)足SKIPIF1<0,則下列說(shuō)法正確的是()A.當(dāng)SKIPIF1<0時(shí),則SKIPIF1<0 B.當(dāng)SKIPIF1<0時(shí),則SKIPIF1<0C.當(dāng)SKIPIF1<0時(shí),則SKIPIF1<0 D.當(dāng)SKIPIF1<0時(shí),則SKIPIF1<0第II卷(非選擇題)二、解答題6.已知數(shù)列SKIPIF1<0滿(mǎn)足SKIPIF1<0,SKIPIF1<0.(1)證明:數(shù)列SKIPIF1<0為等差數(shù)列;(2)設(shè)SKIPIF1<0,證明:SKIPIF1<0.7.已知數(shù)列SKIPIF1<0的前n項(xiàng)和為SKIPIF1<0,對(duì)任意正整數(shù)n,點(diǎn)SKIPIF1<0都在函數(shù)SKIPIF1<0的圖象上,且SKIPIF1<0在點(diǎn)SKIPIF1<0處的切線(xiàn)的斜率為SKIPIF1<0.(1)求數(shù)列SKIPIF1<0的通項(xiàng)公式;(2)若SKIPIF1<0,求證:SKIPIF1<0.8.已知等差數(shù)列SKIPIF1<0的前n項(xiàng)和為SKIPIF1<0,且SKIPIF1<0,又SKIPIF1<0.SKIPIF1<0求數(shù)列SKIPIF1<0的通項(xiàng)公式;SKIPIF1<0若數(shù)列SKIPIF1<0滿(mǎn)足SKIPIF1<0,求證:數(shù)列SKIPIF1<0的前n項(xiàng)和SKIPIF1<0.【答案】(1)SKIPIF1<0(2)證明見(jiàn)解析9.已知等差數(shù)列SKIPIF1<0滿(mǎn)足SKIPIF1<0,SKIPIF1<0,SKIPIF1<0的前n項(xiàng)和為SKIPIF1<0.(1)求SKIPIF1<0及SKIPIF1<0;(2)記SKIPIF1<0,求證:SKIPIF1<0.10.公差不為0的等差數(shù)列SKIPIF1<0的前SKIPIF1<0項(xiàng)和為SKIPIF1<0,若SKIPIF1<0,SKIPIF1<0,SKIPIF1<0,SKIPIF1<0成等比.(1)求數(shù)列SKIPIF1<0的通項(xiàng)公式;(2)設(shè)SKIPIF1<0,證明對(duì)任意的SKIPIF1<0,SKIPIF1<0恒成立.11.已知數(shù)列{an}的前n項(xiàng)和為SnSKIPIF1<0(n∈N*),且a1=2.?dāng)?shù)列{bn}滿(mǎn)足b1=0,b2=2,SKIPIF1<0,n=2,3,….(Ⅰ)求數(shù)列{an}的通項(xiàng)公式;(Ⅱ)求數(shù)列{bn}的通項(xiàng)公式;(Ⅲ)證明:對(duì)于n∈N*,SKIPIF1<0.12.已知函數(shù)SKIPIF1<0的導(dǎo)函數(shù)SKIPIF1<0,數(shù)列SKIPIF1<0的前SKIPIF1<0項(xiàng)和為SKIPIF1<0,點(diǎn)SKIPIF1<0均在函數(shù)SKIPIF1<0的圖象上.若SKIPIF1<0(1)當(dāng)SKIPIF1<0時(shí),試比較SKIPIF1<0與SKIPIF1<0的大??;(2)記SKIPIF1<0試證SKIPIF1<0.13.已知數(shù)列SKIPIF1<0滿(mǎn)足SKIPIF1<0.⑴求SKIPIF1<0;⑵求數(shù)列SKIPIF1<0的通項(xiàng)公式;⑶證明:SKIPIF1<014.?dāng)?shù)列SKIPIF1<0滿(mǎn)足:SKIPIF1<0;數(shù)列SKIPIF1<0滿(mǎn)足:SKIPIF1<0,且SKIPIF1<0.(1)求數(shù)列SKIPIF1<0和SKIPIF1<0的通項(xiàng)公式;(2)設(shè)SKIPIF1<0,證明:SKIPIF1<0;(3)設(shè)SKIPIF1<0,證明:SKIPIF1<0.15.在下列條件:①數(shù)列SKIPIF1<0的任意相鄰兩項(xiàng)均不相等,且數(shù)列SKIPIF1<0為常數(shù)列,②SKIPIF1<0,③SKIPIF1<0中,任選一個(gè),補(bǔ)充在橫線(xiàn)上,并回答下面問(wèn)題.已知數(shù)列SKIPIF1<0的前n項(xiàng)和為SKIPIF1<0,___________.(1)求數(shù)列SKIPIF1<0的通項(xiàng)公式SKIPIF1<0和前n項(xiàng)和SKIPIF1<0;(2)設(shè)SKIPIF1<0,數(shù)列SKIPIF1<0的前n項(xiàng)和記為SKIPIF1<0,證明:SKIPIF1<0.16.已知各項(xiàng)均為正數(shù)的數(shù)列SKIPIF1<0的前SKIPIF1<0項(xiàng)和滿(mǎn)足SKIPIF1<0,且SKIPIF1<0,SKIPIF1<0.(1)求SKIPIF1<0的通項(xiàng)公式;(2)設(shè)數(shù)列SKIPIF1<0滿(mǎn)足SKIPIF1<0,并記SKIPIF1<0為SKIPIF1<0的前SKIPIF1<0項(xiàng)和,求證:SKIPIF1<0,SKIPIF1<0.17.已知數(shù)列SKIPIF1<0中,SKIPIF1<0,(1)求SKIPIF1<0的通項(xiàng)公式;(2)設(shè)SKIPIF1<0,SKIPIF1<0,求證:SKIPIF1<018.?dāng)?shù)列SKIPIF1<0滿(mǎn)足SKIPIF1<0,SKIPIF1<0是SKIPIF1<0的前n項(xiàng)的和,SKIPIF1<0.(1)求SKIPIF1<0;(2)證明:SKIPIF1<0.19.已知各項(xiàng)均為正數(shù)的數(shù)列SKIPIF1<0的前n項(xiàng)和為SKIPIF1<0,且SKIPIF1<0,(1)求證:SKIPIF1<0;(2)求證:SKIPIF1<0.20.已知數(shù)列SKIPIF1<0的首項(xiàng)SKIPIF1<0,SKIPIF1<0,SKIPIF1<0、SKIPIF1<0、SKIPIF1<0.(1)證明:對(duì)任意的SKIPIF1<0,SKIPIF1<0,SKIPIF1<0、SKIPIF1<0、SKIPIF1<0;(2)證明:SKIPIF1<0.21.已知數(shù)列SKIPIF1<0滿(mǎn)足SKIPIF1<0,SKIPIF1<0.(1)證明:數(shù)列SKIPIF1<0是等差數(shù)列;(2)令SKIPIF1<0,證明:SKIPIF1<0.22.已知正項(xiàng)數(shù)列SKIPIF1<0的前SKIPIF1<0項(xiàng)和為SKIPIF1<0,且SKIPIF1<0.(1)求數(shù)列SKIPIF1<0的通項(xiàng)公式;(2)記SKIPIF1<0,證明:當(dāng)SKIPIF1<0時(shí),SKIPIF1<0.23.已知數(shù)列SKIPIF1<0的前n項(xiàng)和為SKIPIF1<0,若SKIPIF1<0.(1)求SKIPIF1<0通項(xiàng)公式;(2)若SKIPIF1<0,SKIPIF1<0為數(shù)列SKIPIF1<0的前n項(xiàng)和,求證:SKIPIF1<0.24.已知數(shù)列SKIPIF1<0滿(mǎn)足SKIPIF1<0,SKIPIF1<0,SKIPIF1<0.(1)設(shè)SKIPIF1<0,求證:數(shù)列SKIPIF1<0是等比數(shù)列;(2)設(shè)數(shù)列SKIPIF1<0的前SKIPIF1<0項(xiàng)和為SKIPIF1<0,求證:SKIPIF1<0,SKIPIF1<0.25.已知數(shù)列SKIPIF1<0滿(mǎn)足SKIPIF1<0.(1)求數(shù)列SKIPIF1<0的通項(xiàng)公式;(2)求證:SKIPIF1<0.26.已知數(shù)列SKIPIF1<0的前n項(xiàng)和為SKIPIF1<0,SKIPIF1<0,SKIPIF1<0.(1)求證SKIPIF1<0為等比數(shù)列;(2)求證:SKIPIF1<0.27.已知數(shù)列SKIPIF1<0的前SKIPIF1<0項(xiàng)和為SKIPIF1<0,SKIPIF1<0,數(shù)列SKIPIF1<0是公差為SKIPIF1<0的等差數(shù)列.(Ⅰ)求數(shù)列SKIPIF1<0的通項(xiàng)公式;(Ⅱ)設(shè)SKIPIF1<0,求證:對(duì)于任意的SKIPIF1<0,SKIPIF1<0.28.已知數(shù)列SKIPIF1<0滿(mǎn)足SKIPIF1<0,SKIPIF1<0,SKIPIF1<0,SKIPIF1<0.(1)(i)證明:數(shù)列SKIPIF1<0是等差數(shù)列;(ii)求數(shù)列SKIPIF1<0的通項(xiàng)公式;(2)記SKIPIF1<0,SKIPIF1<0,SKIPIF1<0,證明:當(dāng)SKIPIF1<0時(shí),SKIPIF1<0.29.已知數(shù)列SKIPIF1<0滿(mǎn)足SKIPIF1<0,SKIPIF1<0,數(shù)列SKIPIF1<0是公比為正數(shù)的等比數(shù)列,SKIPIF1<0,且SKIPIF1<0,SKIPIF1<0,8成等差數(shù)列.(1)求數(shù)列SKIPIF1<0,SKIPIF1<0的通項(xiàng)公式;(2)若數(shù)列SKIPIF1<0滿(mǎn)足SKIPIF1<0,求數(shù)列SKIPIF1<0的前SKIPIF1<0項(xiàng)和SKIPIF1<0.(3)若數(shù)列SKIPIF1<0滿(mǎn)足SKIPIF1<0,求證:SKIPIF1<0.30.已知數(shù)列SKIPIF1<0的首項(xiàng)SKIPIF1<0,其前SKIPIF1<0項(xiàng)和為SKIPIF1<0,且滿(mǎn)足SKIPIF1<0,,其中SKIPIF1<0.(1)求數(shù)列SKIPIF1<0的通項(xiàng)公式;(2)證明:SKIPIF1<0.31.已知數(shù)列SKIPIF1<0滿(mǎn)足SKIPIF1<0,SKIPIF1<0的前SKIPIF1<0項(xiàng)和SKIPIF1<0滿(mǎn)足SKIPIF1<0.(1)求數(shù)列SKIPIF1<0的通項(xiàng)公式;(2)記數(shù)列SKIPIF1<0的前SKIPIF1<0項(xiàng)和為SKIPIF1<0,證明:SKIPIF1<0.32.已知數(shù)列SKIPIF1<0,SKIPIF1<0滿(mǎn)足SKIPIF1<0,SKIPIF1<0(1)若SKIPIF1<0,求證數(shù)列SKIPIF1<0是等差數(shù)列,并求數(shù)列SKIPIF1<0的通項(xiàng)公式:(2)若SKIPIF1<0,(i)求證:SKIPIF1<0;(ii)SKIPIF1<033.已知數(shù)列SKIPIF1<0滿(mǎn)足SKIPIF1<0,SKIPIF1<0,(1)求SKIPIF1<0;(2)若數(shù)列SKIPIF1<0滿(mǎn)足SKIPIF1<0,SKIPIF1<0,求證:SKIPIF1<0.34.設(shè)等差數(shù)列SKIPIF1<0的前SKIPIF1<0項(xiàng)和為SKIPIF1<0,SKIPIF1<0.(1)求SKIPIF1<0與SKIPIF1<0;(2)設(shè)SKIPIF1<0,證明:SKIPIF1<0.35.已知數(shù)列SKIPIF1<0滿(mǎn)足:SKIPIF1<0,SKIPIF1<0,SKIPIF1<0.(1)求證SKIPIF1<0是等差數(shù)列并求SKIPIF1<0;(2)求數(shù)列SKIPIF1<0的前SKIPIF1<0項(xiàng)和SKIPIF1<0;(3)求證:SKIPIF1<0.36.已知數(shù)列SKIPIF1<0滿(mǎn)足SKIPIF1<0,SKIPIF1<0(1)求證:SKIPIF1<0是等比數(shù)列;并寫(xiě)出SKIPIF1<0的通項(xiàng)公式(2)求證:對(duì)任意SKIPIF1<0,有SKIPIF1<037.已知SKIPIF1<0是正項(xiàng)等比數(shù)列SKIPIF1<0的前n項(xiàng)和,且SKIPIF1<0,SKIPIF1<0是SKIPIF1<0,SKIPIF1<0的等差中項(xiàng).(1)求數(shù)列SKIPIF1<0的通項(xiàng)公式;(2)求證:SKIPIF1<0.38.已知數(shù)列SKIPIF1<0滿(mǎn)足SKIPIF1<0,前SKIPIF1<0項(xiàng)和SKIPIF1<0滿(mǎn)足SKIPIF1<0是正項(xiàng)等比數(shù)列,且SKIPIF1<0是SKIPIF1<0和SKIPIF1<0的等比中項(xiàng).(1)求數(shù)列SKIPIF1<0和SKIPIF1<0的通項(xiàng)公式;(2)求證:SKIPIF1<0.39.已知各項(xiàng)均為正數(shù)的數(shù)列SKIPIF1<0滿(mǎn)足:SKIPIF1<0,SKIPIF1<0.(1)求數(shù)列SKIPIF1<0的通項(xiàng)公式;(2)若數(shù)列SKIPIF1<0滿(mǎn)足SKIPIF1<0,SKIPIF1<0,求SKIPIF1<0;(3)若數(shù)列SKIPIF1<0滿(mǎn)足SKIPIF1<0,SKIPIF1<0,求證:SKIPIF1<0.40.已知數(shù)列SKIPIF1<0的前SKIPIF1<0項(xiàng)和為SKIPIF1<0,且SKIPIF1<0.(1)求數(shù)列SKIPIF1<0的通項(xiàng)公式;(2)若數(shù)列SKIPIF1<0的前SKIPIF1<0項(xiàng)和為SKIPIF1<0,證明:SKIPIF1<0.41.已知各項(xiàng)為正數(shù)的數(shù)列SKIPIF1<0滿(mǎn)足:SKIPIF1<0且SKIPIF1<0.(1)證明:數(shù)列SKIPIF1<0為等差數(shù)列.(2)若SKIPIF1<0,證明:對(duì)一切正整數(shù)n,都有SKIPIF1<042.已知數(shù)列SKIPIF1<0滿(mǎn)足:SKIPIF1<0,SKIPIF1<0.(I)求證:數(shù)列SKIPIF1<0是等比數(shù)列;(II)設(shè)SKIPIF1<0的前SKIPIF1<0項(xiàng)和為SKIPIF1<0,求證SKIPIF1<0.43.記SKIPIF1<0為等差數(shù)列SKIPIF1<0的前SKIPIF1<0項(xiàng)和,若SKIPIF1<0,SKIPIF1<0.(1)求SKIPIF1<0和SKIPIF1<0;(2)當(dāng)SKIPIF1<0時(shí),證明:SKIPIF1<0.44.已知正項(xiàng)數(shù)列SKIPIF1<0滿(mǎn)足SKIPIF1<0,SKIPIF1<0.(1)證明:數(shù)列SKIPIF1<0是等比數(shù)列;(2)證明:SKIPIF1<0.45.已知數(shù)列SKIPIF1<0的前n項(xiàng)和記為SKIPIF1<0,且滿(mǎn)足n、SKIPIF1<0、SKIPIF1<0成等差數(shù)列.SKIPIF1<0ⅠSKIPIF1<0求SKIPIF1<0,SKIPIF1<0的值,并證明:數(shù)列SKIPIF1<0是等比數(shù)列;SKIPIF1<0ⅡSKIPIF1<0證明:SKIPIF1<0.46.給定數(shù)列SKIPIF1<0,若滿(mǎn)足SKIPIF1<0SKIPIF1<0且SKIPIF1<0,且對(duì)于任意的SKIPIF1<0,都有SKIPIF1<0,則稱(chēng)數(shù)列SKIPIF1<0為“指數(shù)型數(shù)列”.SKIPIF1<01SKIPIF1<0已知數(shù)列SKIPIF1<0的通項(xiàng)公式SKIPIF1<0,證明:SKIPIF1<0為“指數(shù)型數(shù)列”;SKIPIF1<02SKIPIF1<0若數(shù)列SKIPIF1<0滿(mǎn)足:SKIPIF1<0,SKIPIF1<0;①判斷數(shù)列SKIPIF1<0是否為“指數(shù)型數(shù)列”,若是給出證明,若不是說(shuō)明理由;②若數(shù)列SKIPIF1<0的前SKIPIF1<0項(xiàng)和為SKIPIF1<0,證明:SKIPIF1<0.47.已知數(shù)列SKIPIF1<0中,SKIPIF1<0,其前SKIPIF1<0項(xiàng)的和為SKIPIF1<0,且當(dāng)SKIPIF1<0時(shí),滿(mǎn)足SKIPIF1<0.(1)求證:數(shù)列SKIPIF1<0是等差數(shù)列;(2)證明:SKIPIF1<0.48.已知函數(shù)SKIPIF1<0,數(shù)列SKIPIF1<0中,若SKIPIF1<0,且SKIPIF1<0.(1)求證:數(shù)列SKIPIF1<0是等比數(shù)列;(2)設(shè)數(shù)列SKIPIF1<0的前SKIPIF1<0項(xiàng)和為SKIPIF1<0,求證:SKIPIF1<0.49.設(shè)SKIPIF1<0為數(shù)列SKIPIF1<0的前SKIPIF1<0項(xiàng)和,SKIPIF1<0.(1)求證:數(shù)列SKIPIF1<0是等比數(shù)列;(2)求證:SKIPIF1<0.50.已知數(shù)列SKIPIF1<0中,SKIPIF1<0,其前SKIPIF1<0項(xiàng)和SKIPIF1<0滿(mǎn)足:SKIPIF1<0.(Ⅰ)求數(shù)列SKIPIF1<0的通項(xiàng)公式;(Ⅱ)令SKIPIF1<0,數(shù)列SKIPIF1<0的前SKIPIF1<0項(xiàng)和為SKIPIF1<0,證明:對(duì)于任意的SKIPIF1<0,都有SKIPIF1<0.51.已知數(shù)列SKIPIF1<0的各項(xiàng)均不為零.設(shè)數(shù)列SKIPIF1<0的前SKIPIF1<0項(xiàng)和為SKIPIF1<0,數(shù)列SKIPIF1<0的前SKIPIF1<0項(xiàng)和為SKIPIF1<0,且SKIPIF1<0,SKIPIF1<0.(Ⅰ)求SKIPIF1<0,SKIPIF1<0的值;(Ⅱ)證明數(shù)列SKIPIF1<0是等比數(shù)列,并求SKIPIF1<0的通項(xiàng)公式;(Ⅲ)證明:SKIPIF1<0.52.?dāng)?shù)列SKIPIF1<0前SKIPIF1<0項(xiàng)和為SKIPIF1<0,已知SKIPIF1<0(1)求數(shù)列SKIPIF1<0的通項(xiàng)公式;(2)證明SKIPIF1<0.53.已知數(shù)列SKIPIF1<0滿(mǎn)足SKIPIF1<0,SKIPIF1<0.(1)若SKIPIF1<0為不恒カ0的等差數(shù)列,求SKIPIF1<0;(2)若SKIPIF1<0,證明:SKIPIF1<0.54.?dāng)?shù)列SKIPIF1<0的前n項(xiàng)和為SKIPIF1<0,且滿(mǎn)足SKIPIF1<0,SKIPIF1<0SKIPIF1<0ⅠSKIPIF1<0求通項(xiàng)公式SKIPIF1<0;SKIPIF1<0ⅡSKIPIF1<0記SKIPIF1<0,求證:SKIPIF1<0.55.已知正項(xiàng)數(shù)列SKIPIF1<0滿(mǎn)足SKIPIF1<0.(1)求證:SKIPIF1<0,且當(dāng)SKIPIF1<0時(shí),SKIPIF1<0;(2)求證:SKIPIF1<0.56.已知數(shù)列{an}是等差數(shù)列,數(shù)列{bn}是等比數(shù)列,Sn是數(shù)列{an}的前n項(xiàng)和,a1=b1=1,S2=SKIPIF1<0.(1)若b2是a1,a3的等差中項(xiàng),求數(shù)列{an}與{bn}的通項(xiàng)公式;(2)若an∈N+,數(shù)列{SKIPIF1<0}是公比為9的等比數(shù)列,求證:SKIPIF1<0+SKIPIF1<0+SKIPIF1<0+…+SKIPIF1<0<SKIPIF1<0.57.已知數(shù)列SKIPIF1<0,SKIPIF1<0,二次函數(shù)SKIPIF1<0的對(duì)稱(chēng)軸為SKIPIF1<0.(1)證明:數(shù)列SKIPIF1<0是等差數(shù)列,并求SKIPIF1<0的通項(xiàng)公式;(2)設(shè)SKIPIF1<0,求證:SKIPIF1<0.58.已知數(shù)列SKIPIF1<0的前SKIPIF1<0項(xiàng)和SKIPIF1<0滿(mǎn)足:SKIPIF1<0.(1)數(shù)列SKIPIF1<0的通項(xiàng)公式;(2)設(shè)SKIPIF1<0,且數(shù)列SKIPIF1<0的前SKIPIF1<0項(xiàng)和為SKIPIF1<0,求證:SKIPIF1<0.59.已知數(shù)列SKIPIF1<0滿(mǎn)足SKIPIF1<0,SKIPIF1<0,SKIPIF1<0.(1)求證:數(shù)列SKIPIF1<0是等差數(shù)列;(2)求證:SKIPIF1<0.60.?dāng)?shù)列SKIPIF1<0滿(mǎn)足SKIPIF1<0,SKIPIF1<0.(1)求SKIPIF1<0的值;(2)求數(shù)列SKIPIF1<0的通項(xiàng)公式;(3)設(shè)SKIPIF1<0,求證:SKIPIF1<0.61.設(shè)數(shù)列SKIPIF1<0的前SKIPIF1<0項(xiàng)和為SKIPIF1<0.已知SKIPIF1<0,SKIPIF1<0,SKIPIF1<0.(Ⅰ)求SKIPIF1<0的值;(Ⅱ)求數(shù)列SKIPIF1<0的通項(xiàng)公式;(Ⅲ)證明:對(duì)一切正整數(shù)SKIPIF1<0,有SKIPIF1<0.62.已知函數(shù)SKIPIF1<0,數(shù)列SKIPIF1<0滿(mǎn)足SKIPIF1<0,SKIPIF1<0,SKIPIF1<0.(1)求證:SKIPIF1<0;(2)求證:SKIPIF1<0.63.已知數(shù)列{an}滿(mǎn)足SKIPIF1<0.(Ⅰ)若方程f(x)=x的解稱(chēng)為函數(shù)y=f(x)的不動(dòng)點(diǎn),求an+1=f(an)的不動(dòng)點(diǎn)的值;(Ⅱ)若SKIPIF1<0,求證:數(shù)列{lnbn}是等比數(shù)列,并求數(shù)列{bn}的通項(xiàng).(Ⅲ)當(dāng)任意SKIPIF1<0時(shí),求證:SKIPIF1<0.64.?dāng)?shù)列{an}滿(mǎn)足a(1)求證數(shù)列{a(2)證明:對(duì)一切正整數(shù)n,有1a65.已知數(shù)列SKIPIF1<0滿(mǎn)足條件:SKIPIF1<0,SKIPIF1<0(1)判斷數(shù)列SKIPIF1<0是否為等比數(shù)列;(2)若SKIPIF1<0,令SKIPIF1<0,SKIPIF1<0證明SKIPIF1<066.已知數(shù)列SKIPIF1<0中,SKIPIF1<0,SKIPIF1<0(1)求數(shù)列SKIPIF1<0的通項(xiàng)公式SKIPIF1<0;(2)證明:SKIPIF1<0.67.已知數(shù)列SKIPIF1<0滿(mǎn)足:SKIPIF1<0是公差為1的等差數(shù)列,且SKIPIF1<0(1)求數(shù)列SKIPIF1<0的通項(xiàng)公式SKIPIF1<0;(2)設(shè)SKIPIF1<0,求證:SKIPIF1<068.已知正項(xiàng)數(shù)列SKIPIF1<0滿(mǎn)足:SKIPIF1<0﹣SKIPIF1<0=1,(n∈N+,n≥2),且a1=4.(1)求SKIPIF1<0的通項(xiàng)公式;(2)求證SKIPIF1<0<1(n∈N+)69.已知等差數(shù)列SKIPIF1<0的各項(xiàng)均為正數(shù),SKIPIF1<0=3,前n項(xiàng)和為Sn,SKIPIF1<0是等比數(shù)列,SKIPIF1<0=1,且b2S2=64,b3S3=960.(1)求數(shù)列SKIPIF1<0與SKIPIF1<0的通項(xiàng)公式;(2)求證:SKIPIF1<0對(duì)一切SKIPIF1<0都成立.70.已知正項(xiàng)數(shù)列SKIPIF1<0的前SKIPIF1<0項(xiàng)和為SKIPIF1<0,滿(mǎn)足SKIPIF1<0.(1)求數(shù)列SKIPIF1<0的前SKIPIF1<0項(xiàng)和SKIPIF1<0;(2)記SKIPIF1<0,證明:SKIPIF1<0.71.已知數(shù)列SKIPIF1<0滿(mǎn)足SKIPIF1<0,且點(diǎn)SKIPIF1<0在函數(shù)SKIPIF1<0的圖象上.(1)求證:SKIPIF1<0是等比數(shù)列,并求SKIPIF1<0的通項(xiàng)公式:(2)若SKIPIF1<0,數(shù)列SKIPIF1<0的前n項(xiàng)和為SKIPIF1<0,求證:SKIPIF1<0.72.已知數(shù)列SKIPIF1<0滿(mǎn)足SKIPIF1<0,且當(dāng)SKIPIF1<0時(shí),SKIPIF1<0.(1)求證:數(shù)列SKIPIF1<0是等差數(shù)列,并求數(shù)列SKIPIF1<0的通項(xiàng)公式;(2)記SKIPIF1<0,SKIPIF1<0,證明:當(dāng)SKIPIF1<0時(shí),SKIPIF1<0.73.已知數(shù)列SKIPIF1<0滿(mǎn)足SKIPIF1<0,SKIPIF1<0.(1)證明:數(shù)列SKIPIF1<0為等比數(shù)列,并求數(shù)列SKIPIF1<0的通項(xiàng)公式;(2)求證:SKIPIF1<0.74.已知正項(xiàng)數(shù)列SKIPIF1<0的前SKIPIF1<0項(xiàng)和為SKIPIF1<0,且SKIPIF1<0,SKIPIF1<0.(1)求數(shù)列SKIPIF1<0的通項(xiàng)公式;(2)記SKIPIF1<0,數(shù)列SKIPIF1<0的前SKIPIF1<0項(xiàng)和為SKIPIF1<0,SKIPIF1<0,求證:SKIPIF1<0.75.?dāng)?shù)列SKIPIF1<0滿(mǎn)足SKIPIF1<0,SKIPIF1<0,SKIPIF1<0,SKIPIF1<0.(1)求SKIPIF1<0,SKIPIF1<0及SKIPIF1<0(用SKIPIF1<0表示);(2)設(shè)SKIPIF1<0,求證:SKIPIF1<0;(3)求證:SKIPIF1<0.76.已知SKIPIF1<0是公比SKIPIF1<0的等比數(shù)列,且滿(mǎn)足SKIPIF1<0,SKIPIF1<0,數(shù)列SKIPIF1<0滿(mǎn)足:SKIPIF1<0.(1)求數(shù)列SKIPIF1<0和SKIPIF1<0的通項(xiàng)公式;(2)令SKIPIF1<0,求證:SKIPIF1<0.77.設(shè)數(shù)列SKIPIF1<0的前SKIPIF1<0項(xiàng)和為SKIPIF1<0,且滿(mǎn)足SKIPIF1<0,SKIPIF1<0.(1)求SKIPIF1<0(用SKIPIF1<0表示);(2)求證:當(dāng)SKIPIF1<0時(shí),不等式SKIPIF1<0成立.78.已知函數(shù)SKIPIF1<0,滿(mǎn)足:①對(duì)任意SKIPIF1<0,都有SKIPIF1<0;②對(duì)任意SKIPIF1<0都有SKIPIF1<0.(1)試證明:SKIPIF1<0為SKIPIF1<0上的單調(diào)增函數(shù);(2)求SKIPIF1<0;(3)令SKIPIF1<0,試證明:SKIPIF1<079.已知正項(xiàng)數(shù)列SKIPIF1<0滿(mǎn)足SKIPIF1<0,SKIPIF1<0.(1)試比較SKIPIF1<0與SKIPIF1<0的大小,并說(shuō)明理由;(2)設(shè)數(shù)列SKIPIF1<0的前SKIPIF1<0項(xiàng)和為SKIPIF1<0,證明:當(dāng)SKIPIF1<0時(shí),SKIPIF1<0.80.已知數(shù)列SKIPIF1<0滿(mǎn)足SKIPIF1<0.(1)求數(shù)列SKIPIF1<0的通項(xiàng);(2)設(shè)SKIPIF1<0,若SKIPIF1<0,求證:SKIPIF1<0.81.已知數(shù)列SKIPIF1<0和SKIPIF1<0滿(mǎn)足SKIPIF1<0,且對(duì)任意的SKIPIF1<0,SKIPIF1<0,SKIPIF1<0.(1)求SKIPIF1<0,SKIPIF1<0及數(shù)列SKIPIF1<0的通項(xiàng)公式;(2)記SKIPIF1<0,SKIPIF1<0,求證:SKIPIF1<0,SKIPIF1<0.82.已知數(shù)列SKIPIF1<0的前n項(xiàng)和為SKIPIF1<0,已知SKIPIF1<0,SKIPIF1<0.(1)求數(shù)列SKIPIF1<0的通項(xiàng)公式;(2)證明:SKIPIF1<0.83.正項(xiàng)數(shù)列SKIPIF1<0的前SKIPIF1<0項(xiàng)和為SKIPIF1<0,滿(mǎn)足對(duì)每個(gè)SKIPIF1<0,SKIPIF1<0成等差數(shù)列,且SKIPIF1<0成等比數(shù)列.(1)求SKIPIF1<0的值;(2)求SKIPIF1<0的通項(xiàng)公式;(3)求證:SKIPIF1<084.?dāng)?shù)列SKIPIF1<0,SKIPIF1<0,SKIPIF1<0(1)是否存在常數(shù)SKIPIF1<0,SKIPIF1<0,使得數(shù)列SKIPIF1<0是等比數(shù)列,若存在,求出SKIPIF1<0,SKIPIF1<0的值,若不存在,說(shuō)明理由.(2)設(shè)SKIPIF1<0,SKIPIF1<0,證明:當(dāng)SKIPIF1<0時(shí),SKIPIF1<0.85.已知數(shù)列SKIPIF1<0滿(mǎn)足SKIPIF1<0.(Ⅰ)證明:SKIPIF1<0;(Ⅱ)證明SKIPIF1<0;(Ⅲ)證明:SKIPIF1<0.86.已知數(shù)列SKIPIF1<0的各項(xiàng)均為正數(shù),其前SKIPIF1<0項(xiàng)和為SKIPIF1<0,且滿(mǎn)足SKIPIF1<0,SKIPIF1<0,SKIPIF1<0.(1)求SKIPIF1<0、SKIPIF1<0的值;(2)求數(shù)列SKIPIF1<0的通項(xiàng)公式;(3)證明:對(duì)一切正整數(shù)SKIPIF1<0,有SKIPIF1<0.87.已知數(shù)列SKIPIF1<0滿(mǎn)足SKIPIF1<0,且SKIPIF1<0.(1)證明:SKIPIF1<0;(2)證明:SKIPIF1<0.88.已知數(shù)列SKIPIF1<0、SKIPIF1<0滿(mǎn)足SKIPIF1<0,SKIPIF1<0,SKIPIF1<0,SKIPIF1<0.(Ⅰ)求證:SKIPIF1<0;(Ⅱ)設(shè)數(shù)列SKIPIF1<0的前SKIPIF1<0項(xiàng)和為SKIPIF1<0,求證:SKIPIF1<0;(Ⅲ)設(shè)數(shù)列SKIPIF1<0的前SKIPIF1<0項(xiàng)和為SKIPIF1<0,求證:當(dāng)SKIPIF1<0時(shí),SKIPIF1<0.89.已知數(shù)列SKIPIF1<0滿(mǎn)足SKIPIF1<0,SKIPIF1<0,SKIPIF1<0.(Ⅰ)證明:SKIPIF1<0;(Ⅱ)證明:SKIPIF1<0;(Ⅲ)若SKIPIF1<0,記數(shù)列SKIPIF1<0的前SKIPIF1<0項(xiàng)和為SKIPIF1<0,證明:SKIPIF1<0.90.在數(shù)列SKIPIF1<0中,已知SKIPIF1<0,其中SKIPIF1<0.(1)求SKIPIF1<0的值,并證明:SKIPIF1<0;(2)證明:SKIPIF1<0;(3)設(shè)SKIPIF1<0,求證:SKIPIF1<0.91.已知數(shù)列SKIPIF1<0滿(mǎn)足:SKIPIF1<0,SKIPIF1<0,SKIPIF1<0前SKIPIF1<0項(xiàng)和為SKIPIF1<0的數(shù)列SKIPIF1<0滿(mǎn)足:SKIPIF1<0,SKIPIF1<0,又SKIPIF1<0

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