(本題滿分14分)
已知正項數(shù)列
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滿足:對任意正整數(shù)
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,都有
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成等差數(shù)列,
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成等比數(shù)列,且
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(Ⅰ)求證:數(shù)列
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是等差數(shù)列;
(Ⅱ)求數(shù)列
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的通項公式;
(Ⅲ) 設(shè)
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如果對任意正整數(shù)
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,不等式
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恒成立,求實數(shù)
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的取值范圍.
(1)由已知,得
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①,
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② . 由②得
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③.
將③代入①得,對任意
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,有
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即
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是等差數(shù)列. 4分
(Ⅱ)設(shè)數(shù)列
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的公差為
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,
由
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經(jīng)計算,得
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9分
(Ⅲ)由(1)得
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不等式
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化為
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即
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設(shè)
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,則
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對任意正整數(shù)
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恒成立.
當
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,即
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時,不滿足條件;
當
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,即
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時,滿足條件;
當
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,即
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時,
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的對稱軸為
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,
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關(guān)于
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遞減,
因此,只需
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解得
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綜上,
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14分
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
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