Respuesta :
[tex]\bf \qquad \qquad \textit{sum of a finite arithmetic sequence}\\\\
S_n=\cfrac{n}{2}(a_1+a_n)~
\begin{cases}
n=n^{th}\ term\\
a_1=\textit{first term's value}\\
a_n=n^{th}~value\\
----------\\
a_1=30\\
S_n=-210
\end{cases}
\implies
-210=\cfrac{n}{2}(30+a_n)\\\\\\ -420=n(30+a_n)\implies -420=30n+na_n
\\\\\\
-420-30n=na_n\implies \boxed{\cfrac{-420-30n}{n}=a_n}\\\\
-------------------------------[/tex]
[tex]\bf n^{th}\textit{ term of an arithmetic sequence}\\\\ a_n=a_1+(n-1)d\qquad \begin{cases} n=n^{th}\ term\\ a_1=\textit{first term's value}\\ d=\textit{common difference}\\ ----------\\ a_1=30\\ d=-4 \end{cases} \\\\\\ a_n=30+(n-1)(-4)\implies a_n=30-4n+4 \\\\\\ \boxed{a_n=34-4n}\\\\ -------------------------------[/tex]
[tex]\bf \stackrel{a_n}{34-4n}~~=~~\stackrel{a_n}{\cfrac{-420-30n}{n}}\implies 34n-4n^2=-420-30n \\\\\\ 0=4n^2-64n-420\implies 0=4(n^2-16n-105) \\\\\\ 0=n^2-16n-105\implies 0=(n-21)(n+5)\implies n= \begin{cases} \boxed{21}\\ -5 \end{cases}[/tex]
since the term is just a term unit in the sequence, it cannot be a negative value, thus is not -5.
[tex]\bf n^{th}\textit{ term of an arithmetic sequence}\\\\ a_n=a_1+(n-1)d\qquad \begin{cases} n=n^{th}\ term\\ a_1=\textit{first term's value}\\ d=\textit{common difference}\\ ----------\\ a_1=30\\ d=-4 \end{cases} \\\\\\ a_n=30+(n-1)(-4)\implies a_n=30-4n+4 \\\\\\ \boxed{a_n=34-4n}\\\\ -------------------------------[/tex]
[tex]\bf \stackrel{a_n}{34-4n}~~=~~\stackrel{a_n}{\cfrac{-420-30n}{n}}\implies 34n-4n^2=-420-30n \\\\\\ 0=4n^2-64n-420\implies 0=4(n^2-16n-105) \\\\\\ 0=n^2-16n-105\implies 0=(n-21)(n+5)\implies n= \begin{cases} \boxed{21}\\ -5 \end{cases}[/tex]
since the term is just a term unit in the sequence, it cannot be a negative value, thus is not -5.
