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问题 解答题

已知数列{an}前n项的和为Sn，前n项的积为Tn，且满足Tn=2n（1-n）．

①求a1；

②求证：数列{an}是等比数列；

③是否存在常数a，使得（Sn+1-a）2=（Sn+2-a）（Sn-a）对n∈N+都成立？若存在，求出a，若不存在，说明理由．

答案

（1）∵数列{an}前n项的和为Sn，前n项的积为Tn，且满足Tn=2n（1-n）．

∴a1=T1=21（1-1）=1

（2）证明：∵Tn=2n（1-n）．

∴T（n-1）=2（n-1）（2-n）．

将上面两式相除，

得：an=2[-2（n-1）]．

∴an=

1
4
（n-1）．

∵an+1=

1
4
（n）．

an+1
an
=
1
4

∴数列{an}是等比数列；

（3）∵sn=

1-(
1
4
)n
1-
1
4
=
4
3
-
4(
1
4
)n
3

∴sn+1=

4
3
-
4(
1
4
)n+1
3
，sn+2=
4
3
-
4(
1
4
)n+2
3

∵（Sn+1-a）2=（Sn+2-a）（Sn-a）

∴（Sn+1-a）2=

16[(
1
4
)(2n+2)]
9

而：（Sn+2-a）（Sn-a）=（Sn+2-

4
3
）（Sn-
4
3
）=
16[(
1
4
)(2n+2)]
9

（Sn+1-

4
3
）2=（Sn+2-
4
3
）（Sn-
4
3
）对n∈N+都成立

即：存在常数a=

4
3
，使（Sn+1-a）2=（Sn+2-a）（Sn-a）对n∈N+都成立．

阅读理解
阅读理解。
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1．The word "dog" in Chinese usually _______.
[ ]
A．stands for peace
B．stands for good luck
C．has a negative meaning
D．has a positive meaning
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[ ]
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B．Each person has his own way of life
C．If one works hard,he's sure to succeed
D．Everybody has a time in life to be lucky
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[ ]
A．a tired person
B．a brave man
C．a homeless person
D．an unkind woman
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[ ]
A．"A lucky dog" means "a lucky person" in Western countries.
B．Chinese people prefer dogs to cats.
C．Western people consider cats to be good friends.
D．The rose is the national flower of all the Western countries.
5．What does the writer talk about in the passage?
[ ]
A．Some words have both negative and positive meanings.
B．Different countries have different cultures.
C．The rose means the same in Chinese culture and Western culture.
D．Western culture and Chinese culture have a lot in common.
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