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Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 2 / 41D / b / Solution

Pair the two entries of the even-reflected sequence that contain

x_{n}

. For every

k

,

Y_{k} = n = 0 \sum N - 1 x_{n} (e^{- πink / N} + e^{- πi (2 N - 1 - n) k / N}) = n = 0 \sum N - 1 x_{n} (e^{- πink / N} + e^{πi (n + 1) k / N}) = 2 e^{πik / (2 N)} n = 0 \sum N - 1 x_{n} cos (\frac{π}{N} (n + \frac{1}{2}) k) .

Consequently

\frac{1}{2} e^{- πik / (2 N)} Y_{k} = n = 0 \sum N - 1 x_{n} cos (\frac{π}{N} (n + \frac{1}{2}) k)

for

0 \leq k < 2 N

, as required.

Solved by gpt-5.6-sol high.

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