在一个平面内任意画出六条直线,最多可以把平面分成几个部分?n条呢?
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![在一个平面内任意画出六条直线,最多可以把平面分成几个部分?n条呢?](/uploads/image/z/1984319-71-9.jpg?t=%E5%9C%A8%E4%B8%80%E4%B8%AA%E5%B9%B3%E9%9D%A2%E5%86%85%E4%BB%BB%E6%84%8F%E7%94%BB%E5%87%BA%E5%85%AD%E6%9D%A1%E7%9B%B4%E7%BA%BF%2C%E6%9C%80%E5%A4%9A%E5%8F%AF%E4%BB%A5%E6%8A%8A%E5%B9%B3%E9%9D%A2%E5%88%86%E6%88%90%E5%87%A0%E4%B8%AA%E9%83%A8%E5%88%86%3Fn%E6%9D%A1%E5%91%A2%3F)
在一个平面内任意画出六条直线,最多可以把平面分成几个部分?n条呢?
在一个平面内任意画出六条直线,最多可以把平面分成几个部分?n条呢?
在一个平面内任意画出六条直线,最多可以把平面分成几个部分?n条呢?
对于这类找规律的题目,一般是从最特殊的情况考虑,逐步寻找规律.
为了找到本题的答案,在此我们用圆面表示平面,在上面画直线分割圆面.
1条直线时(如图1):平面(即圆面)最多被分为1+1=2部分;
2条直线时(如图2):平面最多被分为1+1+2=4部分;
3条直线时(如图3):平面最多被分为1+1+2+3=7部分;
4条直线时(如图4):平面最多被分为1+1+2+3+4=11部分;
……
可知:6条直线时:平面最多被分为1+1+2+3+4+5+6=22部分;
n条直线时:平面最多可分为:
1+1+2+3+4+……+n=1+(1+2+3+4+…+n)=1+(1+n)n/2=(2+n+n²)/2(部分).
记画n条直线把平面分成a(n)部分,则画n+1条,a(n+1)=a(n)+n+1;
且a(1)=2;则a(n)=(n^2+n+2)/2;则a(6)=(36+6+2)/2=22;
提供一种简单的求a(n)的方法:
a(1)=2=1+1;
a(2)=a(1)+2;
a(3)=a(2)+3;
.......
a(n)=a(n-1)+n;
两...
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记画n条直线把平面分成a(n)部分,则画n+1条,a(n+1)=a(n)+n+1;
且a(1)=2;则a(n)=(n^2+n+2)/2;则a(6)=(36+6+2)/2=22;
提供一种简单的求a(n)的方法:
a(1)=2=1+1;
a(2)=a(1)+2;
a(3)=a(2)+3;
.......
a(n)=a(n-1)+n;
两式相加得a(n)=1+(1+2+3+4+...+n)=(n^2+n+2)/2
∴ 在同一平面内任意画出6条直线,最多可以把平面分成22个部分
n条为 a(n)=(n^2+n+2)/2
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1条直线可把平面分成二部分,1+1
2条直线最多可把平面分成四部分,1+1+2
3条直线最多可把平面分成七部分,1+1+2+3
4条直线最多可把平面分成十一部分,1+1+2+3+4
5条直线最多可把平面分成十六部分,1+1+2+3+4+5
6条直线最多可把平面分成22部分,1+1+2+3+4+5+6
n条直线最多可把平面分成1+(1+n)n/2部分,...
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1条直线可把平面分成二部分,1+1
2条直线最多可把平面分成四部分,1+1+2
3条直线最多可把平面分成七部分,1+1+2+3
4条直线最多可把平面分成十一部分,1+1+2+3+4
5条直线最多可把平面分成十六部分,1+1+2+3+4+5
6条直线最多可把平面分成22部分,1+1+2+3+4+5+6
n条直线最多可把平面分成1+(1+n)n/2部分,即(n^2+n+2)/2
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a(n)=(n^2+n+2)/2
6条直线:1+1+2+3+4+5+6=22
N条直线:1+1+2+3+4+……+n=1+n
6条直线最多可把平面分成22部分,n条直线最多可把平面分成1+n(n+1)/2
22, 1+n(n+1)/2
(n^2+n+2)*1/2
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