Question

Putin ajutor....()....

Answer 1

$\displaystyle Notam~cu~k~numarul~de~laturi~ale~poligonului. \\ \\ Atunci~suma~unghiurilor~interioare~ale~poligonului~este~ \\ \\ (k-2) \cdot 180^{\circ}. \\ \\ Unghiurile~sunt~120^{\circ},~125^{\circ},~130^{\circ}, \dots,~120^{\circ}+(k-1) \cdot 5^{\circ}. \\ \\ Avem~120^{\circ}+125^{\circ}+130^{\circ}+...+ \left(120^{\circ}+(k-1) \cdot 5^{\circ} \right)=(k-2) \cdot 180^{\circ}.$

$\displaystyle \Leftrightarrow 120^{\circ}+(120^{\circ}+5^{\circ})+(120^{\circ}+2 \cdot 5^{\circ})+...+ \left(120^{\circ}+(k-1) \cdot 5^{\circ} \right)= \\ \\ =(k-2) \cdot 180^{\circ}. \\ \\ \Leftrightarrow k \cdot 120^{\circ}+(1+2+3+...+(k-1)) \cdot 5^{\circ}=(k-2) \cdot 180^{\circ} \\ \\ \Leftrightarrow k \cdot 120^{\circ}+ \frac{(k-1)k}{2} \cdot 5^{\circ}=(k-2) \cdot 180^{\circ} \\ \\ \Leftrightarrow k \cdot 240^{\circ}+(k-1)k \cdot 5^{\circ}=(k-2) \cdot 360^{\circ}$

$\displaystyle \Leftrightarrow k \cdot 240+(k-1)k \cdot 5=(k-2) \cdot 360 ~(Impartim~prin~5) \\ \\ \Leftrightarrow 48k+(k-1)k=72(k-2) \\ \\ \Leftrightarrow 48k+k^2-k=72k-144\\ \\ \Leftrightarrow k^2-25k+144=0 \\ \\ \Leftrightarrow (k-9)(k-16)=0. \\ \\ Deci~k \in \{9,16 \}. \\ \\ \boxed{RASPUNS~(b)}$