Marinada '21

[lang=hr] Neka je $(x_n),n = 1,2,...$ niz definiran sa $x_1 = 9522$ i $$x_1+x_2+...+x_{n-1} = (n^2-1)x_n ,\forall n \geq 2$$ Neka je niz $a_n=x_n+\dfrac{1}{n}S_n, n =1,2,3,...$ gdje je $S_n= x_1 + x_2 +...+x_n$. Odredi sve $n$ za koje je $a_n$ potpuni kvadrat prirodnog broja. Ako je više odgovora napišite ih uzlazno. [/lang] [lang=en] Let $(x_n),n = 1,2,...$ be a sequence defined by $x_1 = 9522$ and $$x_1+x_2+...+x_{n-1} = (n^2-1)x_n ,\forall n \geq 2$$ Let the sequence $a_n=x_n+\dfrac{1}{n}S_n, n =1,2,3,...$ where $S_n= x_1 + x_2 +...+x_n$. Determine the values of $n$ for which the terms of the sequence $a_n$ are perfect squares of an integer. If there is more than one answer, write them in ascending order. [/lang]