Respuesta :
[tex]\bf \qquad \qquad \textit{sum of a finite arithmetic sequence}\\\\
S_n=\cfrac{n}{2}(a_1+a_n)\quad
\begin{cases}
n=n^{th}\ term\\
a_1=\textit{first term's value}\\ a_n=n^{th}~value\\
----------\\
\stackrel{least}{a_1}=128\\\\
\stackrel{greatest}{a_n}=172
\end{cases}[/tex]
[tex]\bf S_n=\cfrac{n}{2}(128~+~172)\implies S_n=\cfrac{n}{2}(300)\implies S_n=150n \\\\\\ \textit{but we also know that sum of all interior angles is }180(n-2) \\\\\\ \stackrel{\textit{sum of the angles}}{S_n}=\stackrel{\textit{sum of the angles}}{180(n-2)}=150n \\\\\\ 180n-360=150n\implies 30n=360\implies n=\cfrac{360}{30}\implies n=12[/tex]
[tex]\bf S_n=\cfrac{n}{2}(128~+~172)\implies S_n=\cfrac{n}{2}(300)\implies S_n=150n \\\\\\ \textit{but we also know that sum of all interior angles is }180(n-2) \\\\\\ \stackrel{\textit{sum of the angles}}{S_n}=\stackrel{\textit{sum of the angles}}{180(n-2)}=150n \\\\\\ 180n-360=150n\implies 30n=360\implies n=\cfrac{360}{30}\implies n=12[/tex]
