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Past exam of the mathematics course of the University of Cambridge / 2025 / ia / Paper 3 / 11A / i / Solution

The ball volume is

V_{n} (R) = \frac{π ^{n /2} R ^{n}}{Γ ( n /2 + 1 )} .

For

n = 2 m

this is

π^{m} R^{2 m} / m!

; for

n = 2 m + 1

it is

2^{2 m + 1} m! π^{m} R^{2 m + 1} / (2 m + 1)!

.

Solved by gpt-5.6-sol high.

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