已知方程tan
2x一

tan x+1=0在x

[0,n

)( n
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N*)內(nèi)所有根的和記為a
n(1)寫出a
n的表達式;(不要求嚴(yán)格的證明)
(2)記S
n = a
1 + a
2 +…+ a
n求S
n;
(3)設(shè)b
n =(kn一5)
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,若對任何n
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N* 都有a
n
b
n,求實數(shù)k的取值范圍.
試題分析:解:( 1)解方程得tanx=
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或
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,當(dāng)n=1時,x=
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或
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,此時
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=
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,
當(dāng)n=2時,x=
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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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+2
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)
依次類推:

=
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+(
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+2
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)+…+[
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+2(n一1)
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],
∴
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=(n
2一
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)

(2)

=(1
2 +2
2 +…+n
2 )

一
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(1+2+…+n)
=
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=
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(3)由

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
得(n2—
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)
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
(kn一5)

,
∴kn

n
2一
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+5 ∵n∈N*,∴k

n+

一

,
設(shè)
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= n+

一

,
易證

在(0,

)上單調(diào)遞減,在(
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,+∞)上單調(diào)遞增
∵n∈N*,

=4,

=4∴n=2,
min =4,
∴k

4
點評:解決的關(guān)鍵是利用數(shù)列的累加法來求解其通項公式,同時能利用分組求和來得到和式,屬于基礎(chǔ)題。
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
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
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