Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 1 / 2G / a

Solution

The

n

[- 1, 1]

T_{n} (x) = cos (n arccos x) .

T_{0} (x) = 1, T_{1} (x) = x, T_{n + 1} (x) = 2 x T_{n} (x) - T_{n - 1} (x) .

Induction therefore shows that every

T_{n}

is a polynomial. Its defining cosine formula also gives

- 1 \leq T_{n} (x) \leq 1 (- 1 \leq x \leq 1) .

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