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[PRONOOBCHICKENHANDSOME]

Cho dãy $a_n:\left\{\begin{matrix} a_1=2 , a_2 =1 \\ a_{n+2}=\frac{n(n+1)a_{n+1}+n^2a_n+5}{n+2}-2 \end{matrix}\right.$
Hãy tìm tất cả các giá trị của n sao cho $a_n \in Z$

Edited by PRONOOBCHICKENHANDSOME, 21-10-2012 - 17:43.

[Noobmath]

Đặt $b_n = na_n \Rightarrow \left\{\begin{matrix} b_1=b_2 = 2 \\ b_{n+2}=nb_{n+1}+nb_n-2n+1\end{matrix}\right.$
Đặt $c_n=b_n -1 \Rightarrow \left\{\begin{matrix} c_1=c_2 = 1 \\ c_{n+2}=nc_{n+1}+nc_n(*)\end{matrix}\right.$
Từ (*) suy ra : $c_{n+2}-(n+1)c_{n+1}=-(c_{n+1}-nc_n)= ... = (-1)^n(c_2-1.c_1)=0$
$\Rightarrow c_{n+2}=(n+1)c_{n+1}=...=(n+1)!$
$\Rightarrow a_n=\frac{(n-1)!+1}{n}$
Suy ra theo định lý Wilson : $a_n \in \mathbb{Z} \Leftrightarrow n \in \mathbb{P}$

Edited by Noobmath, 28-10-2012 - 19:15.