[tex]\it A(x)\cdot A(y) =\begin{pmatrix}\it1&0\\\it x&1\end{pmatrix} \cdot \begin{pmatrix}\it1&0\\\it y&1\end{pmatrix} =\begin{pmatrix}\it1&0\\\it x+y&1\end{pmatrix}\ \ \ \ (*)\\ \\ \\ A(1)\cdot A(4)\cdot A(9)\cdot\ ...\ \cdot A(n^2)=A(55) \stackrel{(*)}{\Longrightarrow}\ 1+4+9+\ ...\ +n^2 =55 \Rightarrow\\ \\ 1^2+2^2+3^2+\ ...\ +n^2=55[/tex]
Verificăm ultima egalitate pentru câteva valori ale lui n.
n=4 ⇒ 1 + 4 + 9 + 16 = 55 ⇒ 30 = 55 (F)
n=5 ⇒ 1 + 4 + 9 + 16 + 25= 55 ⇒ 55 = 55 (A)
Deci, n = 5.
