问题 解答题

对于数列A:a1,a2,…,an,若满足ai∈{0,1}(i=1,2,3,…,n),则称数列A为“0-1数列”。定义变换T,T将"0-1数列"A中原有的每个1都变成0,1,原有的每个0都变成1,0;例如A:1,0,1,则T(A):0,1,1,0,0,1。设A0是"0-1数列",令Ak=T(Ak-1),k=1,2,3,…,

(Ⅰ)若数列A2:1,0,0,1,0,1,1,0,1,0,0,1,求数列A1,A0

(Ⅱ)若数列A0共有10项,则数列A2中连续两项相等的数对至少有多少对?请说明理由;

(Ⅲ)若A0为0,1,记数列Ak中连续两项都是0的数对个数为lk,k=1,2,3,…,求lk关于k的表达式。

答案

解:(Ⅰ)由变换T的定义可得A1:0,1,1,0,0,1;

A0:1,0,1; 

(Ⅱ)数列A0中连续两项相等的数对至少有10对;

证明:对于任意一个“0-1数列”A0,A0中每一个1在A2中对应连续四项1,0,0,1,在A0中每一个0在A2中对应的连续四项为0,1,1,0,

因此,共有10项的“0-1数列”A0中的每一个项在A2中都会对应一个连续相等的数对,

所以A2中至少有10对连续相等的数对。

(Ⅲ) 设Ak中有bk个01数对,

Ak+1中的00数对只能由Ak中的01数对得到,所以

Ak+1中的01数对有两个产生途径:①由Ak中的1得到;②由Ak中00得到,

由变换T的定义及A0:0,1可得Ak中0和1的个数总相等,且共有个,

所以

所以

由A0:0,1可得A1:1,0,0,1,A2:0,1,1,0,1,0,0,1,

所以

当k≥3时,

若k为偶数,

上述各式相加可得

经检验,k=2时,也满足

若k为奇数,

上述各式相加可得

经检验,k=1时,也满足

所以

阅读理解

阅读理解

     One day Mr. Robinson saw a woman in the street with ten children. He was very surprised because

all the children were wearing the same clothes-white hats, blue coats and white pants.

     "Are all those children yours? "he asked the woman.

     "Yes, they are," she answered.

     "Do you always dress them in the same clothes?"asked Mr. Robinson.

     "Yes," answered the mother,"When we had only four children, we dressed them in the same clothes

because we didn't want to lose any of them. It was easy to see our children when they were among (在……中间) others because they were all wearing the same clothes. And now, when we have ten,

we dress them like this because we do not want to take other children home by mistake(错误). When

there are other children among ours, it is easy to see them because their clothes are different."

1. One day __________.

A. Mr. Robinson met a woman teacher with 10 school children

B. Mr. Robinson met 10 children with their mother

C. the mother met Mr. Robinson with 10 school children

D. the mother met Mr. Robinson with his 10 children

2. -Why was Mr. Robinson surprised when he saw the ten children?

    -Because he saw that __________.

A. they all had the same look

B. they were all the mother's children

C. the woman had so many children

D. they were all dressed in the same way

3. When did the mother begin to dress her children in the same clothes?

A. Just after she had the fourth child.

B. Just before the mother had ten children.

C. When she met Mr. Robinson in the street.

D. When she didn't want to take other children home by mistake.

4. According to (根据) the story, we know that __________.

A. all the ten children are boys who are wearing the same clothes

B. all the woman's children are girls who are in the same clothes

C. the children's clothes had the same colour

D. the ten children are both the woman's and Mr. Robinson's children

5. When the woman has ten children, she dresses them in the same clothes because __________.

A. the mother wants to show it to Mr. Robinson

B. she doesn't want to take other children by mistake

C. she wants to take them to walk in the street

D. it's very easy to find them in the street

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