问题 解答题

在数列{an},{bn}中,对任何正整数n都有:a1b1+a2b2+a3b3+…+an-1bn-1+anbn=(n-1)•2n+1

(1)若数列{bn}是首项为1和公比为2的等比数列,求数列{an},{bn}的通项公式;

(2)若数列{an}是首项为a1,公差为d等差数列(a1•d≠0),求数列{bn}的通项公式;

(3)在(2)的条件下,判断数列{bn}是否为等比数列?并说明理由.

答案

(1)因为a1b1+a2b2+a3b3+…+an-1bn-1+anbn=(n-1)•2n+1.

所以a1b1+a2b2+a3b3+…+an-1bn-1=(n-2)•2n-1+1.

两式相减anbn=(2n-2-n+2)•2n-1=n•2n-1

因为{bn}数列是首项为1,公比为2的等比数列,则bn=2n-1

所以an=n;

(2)数列{an}是首项为a1,公差为d等差数列(a1•d≠0),an=a1+(n-1)d,

由(1)可知anbn=n•2n-1,所以bn=

n•2n-1
a1+(n-)d

数列{bn}的通项公式:bn=

n•2n-1
a1+(n-)d

(3){an}是等差数列 anbn=(2n-2-n+2)•2n-1=n•2n-1

所以 an=

n•2n-1
bn

an-1=

(n-1)•2n-2
bn-1

an-2=

(n-2)•2n-3
bn-2

{an}是等差数列 2an-1=an-2+an

(n-1)•2n-2
bn-1
=
n•2n-1
bn
+
(n-2)•2n-3
bn-2
,即
4(n-1)
bn-1
=
4n
bn
+
n-2
bn-2

若{bn}是等比数列,则bn-12=bn-2•bn,上式不满足bn-12=bn-2•bn,所以不成立

所以数列{bn}不是等比数列.

阅读理解

阅读理解。

     Rather than talk about how he feels, often a man would express his love by buying the woman

cosmetics (化妆品), or bringing home a flower, or giving up the control of the remote (遥控器).

     And when men do talk, they'd prefer to talk about actions rather than emotions. For example, a lot of men would choose to express their long-range faith in a relationship by talking about next summer's

vacation plans, not by starting a conversation about undying love.

     Men just prefer to say their plans with plane tickets, rather than poetry. It's one of the reasons why men are more comfortable talking at work (the practical world) than they are at home (the castle that emotion

built). But you can bring out his great communicator by making him feel more like he's operating in the

work mode, even when the topic at hand is your love life.

     When men talk less and women want more, the conflict can escalate. Like when a policeman is

questioning an unwilling witness, more silence equals more questions. A full 65 percent of men we

surveyed recently told us that they don't want their partners to ask them more questions about themselves.

     It's clear that some men are just tired of feeling like they're on the witness stand. They're not necessarily hiding anything; many men simply prefer not to have to tell confusing feelings that they may not even

understand themselves.

     An age-old method can make things better: back off a little, give him room to operate in a conversation, and he's more likely to open up.

1. A man is likely to show his concern for woman by _____.

a. buying her make-up     b. offering her flowers

c. talking about feelings   d. handing over the remote

A. a, b, c  

B. a, b, d  

C. b, c, d  

D. a, c, d

2. According to the author, men feel more comfortable when talking _____.

A. actions  

B. feelings

C. long-term friendship  

D. conversations

3. How can women communicate more successfully with men?

A. They may give men a surprise by buying plane tickets.

B. They can ask men if they have a great time during working.

C. They might create a warm home environment first of all.

D. They can give men the feeling that they are talking at work.

4. The underlined word "escalate" has the similar meaning to _____.

A. decrease  

B. disappear  

C. change  

D. increase

5. Why are men unwilling to be asked about themselves?

A. They have more private affairs than women.

B. They don't want to be seen as criminals.

C. They don't want to be involved in confusing feelings.

D. They have worse ability of understanding themselves.

6. From the text it can be inferred that _____.

A. women are better at writing poems about emotions than men

B. it is natural for women to be fond of finding the faults of men

C. men are more interested in learning more about their partners

D. men prefer to pour out their hearts in a forgiving environment

单项选择题