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Exercice 1: Trouver le terme général de la suite définie par x0=3x_0=3x0=3 et x1=4x_1=4x1=4 et : xn+1=xn−12−nxnx_{n+1}=x_{n-1}^2-nx_nxn+1=xn−12−nxn pour tout n∈Nn\in \mathbb{N}n∈N.
xn=n+3x_n=n+3xn=n+3 par récurrence forte. xn+1=(n+2)2−n(n+3)=n+4x_{n+1}=(n+2)^2-n(n+3)=n+4xn+1=(n+2)2−n(n+3)=n+4 CQFDCQFDCQFD Je n ai pas d exercice à proposer
Exercice2\bold { Exercice 2}Exercice2 Soit TnT_nTn le n-ieme nombre triangulaire ( Tn=n(n+1)2T_n = \frac{n(n+1)}{2}Tn=2n(n+1) ) et SnS_nSn le n-ieme nombre carré ( Sn=n2 S_n = n^2 Sn=n2 ) Trouvez le terme general de An=A_n = An= Sn+Tn+1Sn−1+Tn...S1+T2 \sqrt{S_{n}+T_{n+1}\sqrt{S_{n-1}+T_n\sqrt{...\sqrt{S_1 + T_2}}}} √Sn+Tn+1√Sn−1+Tn√...√S1+T2
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