montrer que l′ensemble des suites reelle qui verifie la relation ∀n∈N aU_(n+2) +bU_(n+1) +cU_n =0 (1) est un espace vectoriel de dimension 2 et determiner une base


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Question Number 145522 by ArielVyny last updated on 05/Jul/21

$m o n t r e r q u e l^{'} e n s e m b l e d e s s u i t e s r e e l l e q u i$ $v e r i f i e l a r e l a t i o n \forall n \in N$ ${a U}_{n + 2} + {b U}_{n + 1} + {c U}_{n} = 0 (1) e s t u n e s p a c e$ $v e c t o r i e l d e d i m e n s i o n 2$ $e t d e t e r m i n e r u n e b a s e$
	Answered by Olaf_Thorendsen last updated on 05/Jul/21

	$Les 2 suites reelles a_{n} = r_{1}^{n} et b_{n} = r_{2}^{n} ou$ $r_{1} et r_{2} sont les racines du trinome$ ${a x}^{2} + b x + c = 0 forment une famille$ $libre et generatrice . Toute suite U_{n}$ $verifiant la relation (1) est combinaison$ $lineaire de a_{n} et b_{n} (se verifie aisement$ $par recurrence) .$
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