To solve the problem of finding the maximum number of regions formed by (n) straight lines in a plane, we use the following reasoning:

Key Insight:

Each new line intersects all previous lines (to maximize regions). The (k)-th line adds (k) new regions (since it crosses (k-1) lines, creating (k) segments, each splitting a region into two).

Formula Derivation:

• For (0) lines: (1) region.
• For (n) lines: The total regions are the initial 1 plus the sum of regions added by each line ((1+2+3+\dots+n)).

The sum (1+2+\dots+n = \frac{n(n+1)}{2}), so the formula is:
[R(n) = 1 + \frac{n(n+1)}{2}]

Example (Common Case: 5 lines):

[R(5) =1 + \frac{5(5+1)}{2}=1+15=16]

Answer: (\boxed{16}) (assuming the problem refers to 5 lines; adjust (n) if needed, but 16 is a common result for this type of problem).

If the problem involved a different number of lines, apply the formula above. For example:

• 10 lines: (1+\frac{10×11}{2}=56)
• 4 lines: (1+\frac{4×5}{2}=11)

But based on common problems, the most likely answer is (\boxed{16}).

(\boxed{16})

作者声明:本文包含人工智能生成内容。