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问题 解答题
已知数列{an}的前n项和为Sn．且满足Sn=2an-1（n∈N+）
（I）求证：数列{an}是等比数列；
（II）数列{bn}满足bn+1．=an+bnn∈N+．且b1=3．若不等式log2(bn-2)＜
3
16
n2+t对任意n∈N+恒成立，求实数t的取值范围．
答案

（ I）证明：依题意可得Sn+1=2an+1-1…①，Sn=2an-1…②

①-②，得an+1=2an+1-2an

化简得

an+1
an
=2(n∈N*)，

∵a1=2a1-1，

∴a1=1

∴数列{an}是以1为首项，公比为2的等比数列．

（II）由（Ⅰ）可知an=2n-1，因为bn+1=an+bn，n∈N+．且b1=3，

所以bn=an-1+bn-1=an-1+an-2+bn-2=…=an-1+an-2+…+a1+b1

=2n-2+2n-3+…+1+3=2n-1+2，

因为不等式log2(bn-2)＜

3
16
n2+t对任意n∈N+恒成立，

所以log2(2n-1+2-2)＜

3
16
n2+t，

即t＞-

3
16
n2+n-1，对任意n∈N+恒成立，

因为-

3
16
n2+n-1≤
5
16
，且n=3时-
3
16
n2+n-1取得最大值
5
16
．

所以t＞

5
16
．

所以实数t的取值范围：(

5
16
，+∞)．

选择题

I’m ____ heavier than my brother. 

A．a few

B．few

C．a little

D．little
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阅读理解

Can trees talk? Yes, but not in words. Scientists have reason to believe that trees do communicate (交际) with each other. Not long ago, researchers learned some surprising things. First a willow tree attacked in the woods by caterpillars (毛虫) changed the chemistry of its leaves and made them taste so terrible that they got tired of the leaves and stopped eating them. Then even more astonishing, the tree sent out a special smell---a signal (信号) causing its neighbors to change the chemistry of their own leaves and make them less tasty.

Communication, of course, doesn’t need to be in words. We can talk to each other by smiling, raising our shoulders and moving our hands. We know that birds and animals use a whole vocabulary of songs, sounds, and movements. Bees dance their signals, flying in certain patterns that tell other bees where to find nectar (花蜜) for honey. So why shouldn’t trees have ways of sending message?

小题1: It can be concluded from the passage that caterpillars do not feed on leaves that ______.

A．are lying on the ground

B．have an unpleasant taste

C．bees don’t like

D．have an unfamiliar shape小题2:According to the passage, the willow tree was able to communicate with other trees by ______.

　    

A．waving its branches

B．giving off a special smell

C．dropping its leaves

D．changing the colour of its trunk小题3:According to this passage, bees communicate by ______.

A．making special movement

B．touching one another

C．smelling one another

D．making unusual sound
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