The text in the image is in Chinese and translates as follows:

"(1) There are $n$ straight lines on a plane, none of which are parallel, and no three lines intersect at the same point. How many regions do these lines divide the plane into?"

This is a classic problem in combinatorics. The formula for the number of regions $R$ formed by $n$ lines in a plane, given that no two lines are parallel and no three lines are concurrent, is:

$R(n) = \frac{n(n + 1)}{2} + 1$

Explanation:

• For $n = 0$, there is 1 region (the entire plane).
• For $n = 1$, the plane is divided into 2 regions by one line.
• For $n = 2$, the plane is divided into 4 regions by two intersecting lines.
• And so on...

Would you like me to solve a specific case of this problem or explore related questions?

Here are some related questions you might find useful:

1. How many regions does the plane have if there are 5 lines?
2. What happens if two lines are parallel? How does that affect the number of regions?
3. How can this formula be derived using combinatorics?
4. What if three or more lines intersect at a single point—how do we adjust the formula?
5. How many intersections are created by $n$ lines on the plane?

Tip: Try drawing diagrams with different numbers of lines to visualize how the regions are formed!