已知等差數(shù)列
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823212729970378.png)
的首項(xiàng)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823212729986351.png)
,公差
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823212730298362.png)
,且第2項(xiàng)、第5項(xiàng)、第14項(xiàng)分別是等比數(shù)列
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823212730313368.png)
的第2項(xiàng)、第3項(xiàng)、第4項(xiàng)。
①求數(shù)列
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823212729970378.png)
與
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823212730313368.png)
的通項(xiàng)公式;
②設(shè)數(shù)列
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823212730360373.png)
對(duì)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823212730376407.png)
均有
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823212730594761.png)
成立,求
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823212730610593.png)
+
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823212730625418.png)
(1)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823212730937795.png)
(2)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823212730953469.png)
(1)利用等差數(shù)列和等比數(shù)列的概念及性質(zhì)易求兩個(gè)數(shù)列的通項(xiàng)公式;(2)先求出數(shù)列
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823212730969522.png)
的通項(xiàng)公式,然后再利用分組求和及等比數(shù)列的前N項(xiàng)和公式求出數(shù)列的前2011項(xiàng)和
(1)由已知有
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/201408232127309841044.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/201408232127315931276.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823212732092471.png)
,解得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823212732185416.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823212733106929.png)
…………………3分
又
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823212733121654.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823212733121668.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823212733137195.png)
數(shù)列
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823212733277568.png)
的公比為3.
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823212730937795.png)
…………………5分
(2)由
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823212733480504.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823212733589557.png)
…
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823212733683764.png)
得,當(dāng)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823212733714435.png)
時(shí),
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823212733480504.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823212733589557.png)
…
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823212733792755.png)
兩式相減得,當(dāng)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823212733714435.png)
時(shí),
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823212733823927.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/201408232127338391065.png)
…………………8分
又
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823212733870357.png)
時(shí),
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823212733886878.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/201408232127339171322.png)
,…………………10分
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823212733933722.png)
=3+
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823212733948894.png)
3-3+
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823212730953469.png)
=
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823212730953469.png)
。…………………12分
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