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Let \((u_n)\) be the sequence defined by \(u_0 = 2\) and for all \(n \in \mathbb{N}\), \(u_{n+1} = 4u_n - 9\).
Let \((v_n)\) be the sequence defined for all \(n \in \mathbb{N}\) by \(v_n = u_n - 3\).
Calculate the first three terms of the sequence \((u_n)\).
Show that the sequence \((v_n)\) is geometric. Specify its common ratio and its first term.
Deduce the expression of \(v_n\) in terms of \(n\).
Deduce the expression of \(u_n\) in terms of \(n\).
Calculate \(u_{10}\).
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