已知數(shù)列
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滿足
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,

.
(1)求
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的值,由此猜測(cè)
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的通項(xiàng)公式,并證明你的結(jié)論;
(2)證明:
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.
(1)猜想
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,證明詳見(jiàn)解析;(2)證明詳見(jiàn)解析.
試題分析:(1)根據(jù)遞推關(guān)系,依次附值
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即可得到
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的取值,進(jìn)而作出猜想
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,然后再用數(shù)學(xué)歸納法證明即可;(2)先化簡(jiǎn)
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,進(jìn)而采用放縮法得到
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,進(jìn)而將
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取1,2,3,……,
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時(shí)的不等式相乘即可證明不等式
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,然后構(gòu)造函數(shù)
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,確定該函數(shù)在區(qū)間
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上的單調(diào)性,進(jìn)而得到
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在
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恒成立,從而可得
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即
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,問(wèn)題得以證明.
(1)令
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可知
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,
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,
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猜想

,下用數(shù)學(xué)歸納法證明.
(1)
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時(shí),顯然成立;
(2)假設(shè)
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時(shí),命題成立.即
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.
當(dāng)
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時(shí),由題可知
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.
故
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時(shí),命題也成立.
由(1)(2)可知,
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.
(2)證明:∵
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∴
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由于
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,可令函數(shù)
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,則
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,令
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,得
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,給定區(qū)間
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,則有

,則函數(shù)
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在
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上單調(diào)遞減,∴
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,即
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在
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恒成立,又
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,則有
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,即
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所以
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.
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
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
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
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
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
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
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
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
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
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
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
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
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
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
的前

項(xiàng)的和,其中
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
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
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
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
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若數(shù)列

滿足
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(
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為常數(shù)),則稱數(shù)列

為“等比和數(shù)列” ,

稱為公比和。已知數(shù)列

是以3為公比和的等比和數(shù)列,其中

,

,則
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A.1 | B.2 | C. | D. |
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題型:單選題
設(shè)
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為等差數(shù)列
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的前
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項(xiàng)和,若
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,公差
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,
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,則
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