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Fie sirul ([tex] a_{n} [/tex]) , [tex] a_{1} =5 ~;~ a_{2} =17 \\ si~a _{k+2} =5 *a_{k+1} +4 *a_{k} ,k \geq 1.Sa~se~dem.~ca~ a_{n} = 4^{n} +1,n \geq 1.[/tex]

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Matepentrutoti
Recurenta data este de ordin 2.
[tex]a_{k+2}-5a_{k+1}+4a_k=0\\ Ecuatia~caracteristica~asociata~este:\\ r^2-5r+4=0\\ \triangle=25-16=9\\ x_{1/2}= \frac{5\pm3}{2} \\ x_1=4\\ x_2=1\\ Sirul~a_n~are~forma:\\ a_n=A\cdot x_1^n+B\cdot x_2^n\\ a_n=A\cdot 4^n+B\cdot 1^n\\ ~Pentru~a ~determina~A~si~B~ne~folosim~de ~a_1=5~si~a_2=17.\\ 4A+B=5\\ 16A+B=17\\ A=B=1\\ In~final,\\ a_n=4^n+1.[/tex]

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