試題分析:(Ⅰ)根據(jù)題意可得當
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824031344511515.png)
時,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824031344542545.png)
成等差數(shù)列,當
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824031344558515.png)
時,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824031344573532.png)
,可見由
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824031344589423.png)
得出前
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824031344620323.png)
項成等差數(shù)列,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824031344636333.png)
項以后奇數(shù)項為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824031344651266.png)
,偶數(shù)項為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824031344667206.png)
,這樣結(jié)合等差數(shù)列的前
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824031344698297.png)
項公式就可求出
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824031344698412.png)
;(Ⅱ)以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824031344714256.png)
和
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824031344651266.png)
為界對
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824031344761315.png)
進行分類討論,當
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824031344776547.png)
時,顯然成立;當
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824031344792451.png)
時,由題中所給數(shù)列的遞推關(guān)系
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824031344807717.png)
,不難得到
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824031344823850.png)
;當
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824031344839431.png)
時,得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824031344854621.png)
,可轉(zhuǎn)化為當
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824031344792451.png)
時的情況,命題即可得證; (Ⅲ)由
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824031344464486.png)
可得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824031344885541.png)
,根據(jù)題中遞推關(guān)系可得出
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240313449011280.png)
,進而可得出
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824031344449779.png)
=
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240313449321612.png)
,又
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240313449481024.png)
,由于
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824031344948365.png)
要對
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824031344698297.png)
分奇偶性,故可將相鄰兩整數(shù)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824031344979530.png)
當作一個整體,要證不等式可進行適當放縮
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824031344995867.png)
,要對
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824031344698297.png)
分奇偶性,并結(jié)合數(shù)列求和的知識分別進行證明即可.
試題解析:(Ⅰ)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824031345026535.png)
由題意知數(shù)列
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824031344293481.png)
的前34項成首項為100,公差為-3的等差數(shù)列,從第35項開始,奇數(shù)項均為3,偶數(shù)項均為1,從而
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824031344698412.png)
=
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240313450731458.png)
(3分)
=
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240313450881667.png)
. (5分)
(Ⅱ)證明:①若
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824031344776547.png)
,則題意成立 (6分)
②若
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824031344792451.png)
,此時數(shù)列
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824031344293481.png)
的前若干項滿足
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824031345135539.png)
,即
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824031344807717.png)
.
設(shè)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240313451661149.png)
,則當
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824031345182452.png)
時,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824031344823850.png)
.
從而此時命題成立 (8分)
③若
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824031344839431.png)
,由題意得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824031344854621.png)
,則由②的結(jié)論知此時命題也成立.
綜上所述,原命題成立 (10分)
(Ⅲ)當
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824031344464486.png)
時,因為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240313449011280.png)
,
所以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824031344449779.png)
=
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240313449321612.png)
(11分)
因為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824031344948365.png)
>0,所以只要證明當
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824031345322411.png)
時不等式成立即可.
而
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240313453382243.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240313453691341.png)
(13分)
①當
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824031345385926.png)
時,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240313454002301.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240313454161398.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240313454311437.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824031345463580.png)
(15分)
②當
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824031345463961.png)
時,由于
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824031344948365.png)
>0,所以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824031345494879.png)
<
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824031345509563.png)
綜上所述,原不等式成立 (16分)