(本小題滿分14分)
已知數(shù)列
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是首項為
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,公差為

的等差數(shù)列,
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是首項為
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,公比為
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的等比數(shù)列,且滿足
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,其中
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.
(Ⅰ)求
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的值;
(Ⅱ)若數(shù)列
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與數(shù)列
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有公共項,將所有公共項按原順序排列后構(gòu)成一個新數(shù)列
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,求數(shù)列
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的通項公式;
(Ⅲ)記(Ⅱ)中數(shù)列
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的前項之和為
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,求證:
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.
解:(Ⅰ)由題設(shè)
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. ………1分
由已知
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,所以
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.又
b>0,所以
a<3. …2分
因為
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,則
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.又
a>0,所以
b>2,從而有

. …3分
因為
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,故
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. …………………4分
(Ⅱ)設(shè)

,即

. ………………5分
因為
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,則
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,所以
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. ………………6分
因為
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,且
b∈N*,所以
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,即
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,且
b=3. ……7分
故

. …………………8分
(Ⅲ)由題設(shè),
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. …………………9分
當(dāng)
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
時,
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,當(dāng)且僅當(dāng)
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時等號成立,所以

. ……………11分
于是

. 12分
因為S
1=3,S
2=9,S
3=21,則



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. …………………14分
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已知數(shù)列
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
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
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
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
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
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