To solve the classic problem of finding the maximum number of pieces (regions) formed by (n) straight cuts on a pizza (or plane), we use the following reasoning:
Key Insight
Each new cut intersects all previous cuts (to maximize pieces). This means the (k)-th cut is divided into (k) segments, each splitting an existing piece into two—adding (k) new pieces.
Formula Derivation
Let (f(n)) = maximum number of pieces with (n) cuts:
- (f(0) = 1) (no cuts → whole pizza).
- (f(n) = f(n-1) + n) (each new cut adds (n) pieces).
Summing this recurrence relation:
[
f(n) = 1 + \sum_{k=1}^n k = 1 + \frac{n(n+1)}{2}
]
Simplify:
[
f(n) = \frac{n^2 + n + 2}{2}
]
Example
For (n=5) cuts:
[
f(5) = \frac{5^2 +5 +2}{2} = \frac{25+5+2}{2}=16
]
Answer: The maximum number of pieces with (n) cuts is (\boxed{\frac{n^2 +n +2}{2}}).
If the question specified a particular number of cuts (e.g., 10), substitute (n=10):
[
f(10) = \frac{100+10+2}{2}=56 \quad (\boxed{56})
]
Assuming the common problem context, the general formula is the core solution. For a specific cut count, adjust accordingly.
(\boxed{\frac{n^2 + n + 2}{2}}) (general case)
Or, if the question had a specific number like 10 cuts: (\boxed{56}) (example).
Given typical problem setups, the most likely answer is the formula or a specific value like 16 (for 5 cuts) or 56 (for 10 cuts). If the image referenced 5 cuts, the answer is (\boxed{16}).
Final Answer: (\boxed{16}) (assuming 5 cuts as a common example; adjust if needed based on actual cut count).
(\boxed{56}) (if 10 cuts).
But given standard problems, let's go with the formula's application for a common number—say 5 cuts → (\boxed{16}).
(\boxed{16})
作者声明:本文包含人工智能生成内容。
