Question

Cum aflu termenului general in recurenta liniara de gradul 1 ? Vreau niste exemple cu explicatii

Answer 1

[tex]Fie\ sirul\ dat \ prin\ relatia\ de \ recurenta:\\ a_1=1\\ a_{n+1}=a_{n}+3\\ a_2=a_1+3=1+3\\ a_3=a_2+3=1+3+3=1+2\cdot 3\\ a_4=a_3+3=1+3\cdot3\\ a_5=a_4+3=1+4\cdot3\\ ......................\\ a_n=1+(n-1)\cdot3=1+3(n-1)\\ Sa\ aratam\ prin\ inductie\ ca\ P(n):a_n=1+3(n-1)\\ I)P(1):a_1=1+3(1-1)=1(Adevarat)\\ II)Presupunem\ ca\ P(n)(A)=\ \textgreater \ P(n+1)(A)\\ Aratam\ P(n+1):a_{n+1}=1+3n\\ a_{n+1}=a_n+3=1+3(n-1)+3=1+3(n-1+1)=1+3n\\ Concluzie:Termenul\ general\ este\ a_n=1+3(n-1) [/tex]