grandes-ecoles 2021 Q1

grandes-ecoles · France · centrale-maths2__official Proof by induction Prove a general algebraic or analytic statement by induction

For all $n$ in $\mathbb{N}$, determine the degree of $T_n$, then show that $\left(T_k\right)_{0 \leqslant k \leqslant n}$ is a basis of $\mathbb{C}_n[X]$.
The sequence of polynomials $\left(T_n\right)_{n \in \mathbb{N}}$ is defined by $T_0 = 1, T_1 = X$ and $\forall n \in \mathbb{N}, T_{n+2} = 2X T_{n+1} - T_n$.

For all $n$ in $\mathbb{N}$, determine the degree of $T_n$, then show that $\left(T_k\right)_{0 \leqslant k \leqslant n}$ is a basis of $\mathbb{C}_n[X]$.

The sequence of polynomials $\left(T_n\right)_{n \in \mathbb{N}}$ is defined by $T_0 = 1, T_1 = X$ and $\forall n \in \mathbb{N}, T_{n+2} = 2X T_{n+1} - T_n$.

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