10 La relation de récurrence d'une suite \( u \) est : \( u(n+1)=4-(u(n))^{2} \) et \( u(1)=2 \) a) Calculer \( u(2), u(3) \) et \( u(4) \). b) Reprendre avec \( u(1)=-3 \).
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Pour la suite donnée, calculons les premiers termes. a) À partir de \( u(1) = 2 \): - \( u(2) = 4 - (u(1))^2 = 4 - 2^2 = 4 - 4 = 0 \) - \( u(3) = 4 - (u(2))^2 = 4 - 0^2 = 4 - 0 = 4 \) - \( u(4) = 4 - (u(3))^2 = 4 - 4^2 = 4 - 16 = -12 \) Donc, les valeurs sont \( u(2) = 0 \), \( u(3) = 4 \) et \( u(4) = -12 \). b) Maintenant, reprenons avec \( u(1) = -3 \): - \( u(2) = 4 - (u(1))^2 = 4 - (-3)^2 = 4 - 9 = -5 \) - \( u(3) = 4 - (u(2))^2 = 4 - (-5)^2 = 4 - 25 = -21 \) - \( u(4) = 4 - (u(3))^2 = 4 - (-21)^2 = 4 - 441 = -437 \) Les valeurs sont maintenant \( u(2) = -5 \), \( u(3) = -21 \) et \( u(4) = -437 \).