設(shè)
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是數(shù)列
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的前
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項和,對任意
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都有
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成立, (其中
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、
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、
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是常數(shù)).
(1)當
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,
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,
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時,求
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;
(2)當
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,
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,
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時,
①若
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,
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,求數(shù)列
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的通項公式;
②設(shè)數(shù)列
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中任意(不同)兩項之和仍是該數(shù)列中的一項,則稱該數(shù)列是“
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數(shù)列”.
如果
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,試問:是否存在數(shù)列
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為“
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數(shù)列”,使得對任意
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,都有
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,且
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.若存在,求數(shù)列
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的首項
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的所
有取值構(gòu)成的集合;若不存在,說明理由.
(1)
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=
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;(2)①
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;②存在,首項
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的所有取值構(gòu)成的集合為
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.
試題分析:(1)要求
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,大多數(shù)時候要先求
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,本題實質(zhì)就是有關(guān)系式
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,那么我們可以用
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代
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得
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,兩式相減,可得出
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與
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的關(guān)系,本題正好得到數(shù)列
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是等比數(shù)列,故易求得
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和
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;(2) 實質(zhì)上的關(guān)系式是
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,這讓我們聯(lián)想到數(shù)列
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是等差數(shù)列,這里難點就在于證明
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是等差數(shù)列,證明方法是把等式
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中的
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用
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換得到一個式子,兩式相減可得
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,此式中含有常數(shù)
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,故再一次用
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代換此式中的
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,兩式相減可消去

得數(shù)列
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的連續(xù)三項
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的關(guān)系,可證得
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是等差數(shù)列,那么這里①的通項公式易求;對于②這類問題總是假設(shè)存在,然后去求,假設(shè)存在時,可知數(shù)列公差是2,即
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,由于它是“
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數(shù)列”,故任意兩項和還是數(shù)列中的項,即
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,可得
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是偶數(shù),又由
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,得
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,娵
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,從而
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,下面對
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的值一一驗證是否符合已知條件
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,
試題解析:(1)當
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,
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,
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時,由
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得
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①
用
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去代
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得,
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, ②
②—①得,
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,
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,
在①中令
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得,
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,則
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0,∴
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,
∴數(shù)列
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是以首項為1,公比為3的等比數(shù)列,
∴
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=
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(2)當
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,
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,
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時,
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, ③
用
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去代
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得,
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, ④
④—③得,
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, ⑤
用
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去代
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得,
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, ⑥
⑥—⑤得,
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,即
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,
∴數(shù)列
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是等差數(shù)列.∵
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,
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,
∴公差
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,∴
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易知數(shù)列
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是等差數(shù)列,∵
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,∴
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.
又
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是“
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數(shù)列”,得:對任意
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,必存在
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使
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,
得

,故

是偶數(shù),
又由已知,

,故

一方面,當

時,


,對任意

,
都有

另一方面,當

時,

,

,
則

,
取

,則

,不合題意.
當

時,

,

,則


,
當

時,


,

,

,
又

,∴

或

或

或

所以,首項

的所有取值構(gòu)成的集合為

(其他解法,可根據(jù)【解】的評分標準給分)

與

的關(guān)系,求

和

;(2)等差數(shù)列的通項公式,前

項和

.
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科目:高中數(shù)學
來源:不詳
題型:解答題
設(shè)數(shù)列{a
n} 的前n項和為S
n,滿足2S
n=a
n+1﹣2
n+1+1,n∈N
*,且a
1,a
2+5,a
3成等差數(shù)列.
(1)求a
1,a
2,a
3的值;
(2)求證:數(shù)列{a
n+2
n}是等比數(shù)列;
(3)證明:對一切正整數(shù)n,有

+

+…+

<

.
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科目:高中數(shù)學
來源:不詳
題型:解答題
(本小題滿分12分)已知函數(shù)

,當

時取得最小值-4.
(1)求函數(shù)

的解析式;
(2)若等差數(shù)列

前n項和為

,且

,

,求數(shù)列

的前n項和

.
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科目:高中數(shù)學
來源:不詳
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數(shù)列

中,已知

,

時,

.數(shù)列

滿足:

.
(1)證明:

為等差數(shù)列,并求

的通項公式;
(2)記數(shù)列

的前

項和為

,若不等式

成立(

為正整數(shù)).求出所有符合條件的有序?qū)崝?shù)對

.
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已知等差數(shù)列

中,公差

,其前

項和為

,且滿足:

,

.
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
的通項公式;
(Ⅱ)令

,

(

),求

的最大值.
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已知公差不為零的等差數(shù)列

的前3項和

,且

、

、

成等比數(shù)列.
(1)求數(shù)列

的通項公式及前n項的和

;
(2)設(shè)

的前n項和,證明:

;
(3)對(2)問中的

,若

對一切

恒成立,求實數(shù)

的最小值.
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來源:不詳
題型:單選題
在等差數(shù)列

中,

,

,記數(shù)列

的前

項和為

,若

對

恒成立,則正整數(shù)

的最小值為( )
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