To determine the maximum number of regions formed by (n) intersecting circles, we can derive the formula step-by-step:

Key Observations:

• Base Case: For (n=1), there are 2 regions (inside and outside the circle).
• Inductive Step: Each new circle intersects all previous ((n-1)) circles at 2 points each, creating (2(n-1)) intersection points. These points split the new circle into (2(n-1)) arcs, each adding one new region.

Deriving the Formula:

Let (R(n)) be the number of regions for (n) circles:
[R(n) = R(n-1) + 2(n-1)]

Expanding this recurrence relation:
[R(n) = 2 + \sum_{k=1}^{n-1} 2k]

The sum of the first ((n-1)) integers is (\frac{(n-1)n}{2}), so:
[R(n) = 2 + 2 \times \frac{(n-1)n}{2} = n^2 - n + 2]

Answer: (\boxed{n^2 - n + 2})

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