[tex]a_4=7\,,\quad a_{8}=112[/tex]
[tex]\begin{cases}a_4=a_1+3r\\a_8=a_1+7r\end{cases}\\\\\begin{cases}7=a_1+3r\qquad/\cdot(-1)\\112=a_1+7r\end{cases}\\\\\underline{\begin{cases}-7=-a_1-3r\\112=a_1+7r\end{cases}}\\{}\ \ \,105\,=\,4r\qquad/:4\\{}\quad r=26\frac14\\\\ 7=a_1+3\cdot26\frac14\\7=a_1+78\frac34\\a_1=-71\frac34\\\\\\ S_n=\dfrac{2a_1+(n-1)r}2\cdot n\\\\S_{13}=\dfrac{2a_1+12r}2\cdot 13\\\\ S_{13}= \dfrac{2\cdot(-71\frac34)+12\cdot26\frac14}2\cdot 13\\\\ S_{13}= \dfrac{-143\frac12+315}2\cdot 13\\\\S_{13}=\dfrac{171\frac12}2\cdot 13[/tex]
[tex]S_{13}=85\frac34\cdot 13\\\\S_{13}=1\,114\frac34[/tex]
[tex]a_4=a_1\cdot q^3\quad\implies\quad a_1=\dfrac{a_4}{q^3}=\dfrac{7}{q^3}\\\\\\a_8=a_1\cdot q^7\\\\112=\dfrac{7}{q^3}\cdot q^7\\\\112=7 q^4\qquad/:7\\\\q^4=16\\\\{}\ \, q=2\qquad\vee\qquad q=-2\\\\a_1=\frac78\qquad\vee\qquad a_1=-\frac78\\\\\\S_n=a_1\cdot\dfrac{1-q^n}{1-q}\\\\S_{13}=a_1\cdot\dfrac{1-q^{13}}{1-q}\\\\S_{13}=\dfrac78\cdot\dfrac{1-2^{13}}{1-2}\qquad\vee\qquad S_{13}=-\dfrac78\cdot\dfrac{1-(-2)^{13}}{1-(-2)}[/tex]
[tex]S_{13}=\dfrac78\cdot\dfrac{1-8192}{-1}\qquad\vee\qquad S_{13}=-\dfrac78\cdot \dfrac{1-(-8192)}{3}\\\\ S_{13}=\frac78\cdot8191\qquad\qquad\vee\qquad S_{13}=-\frac78\cdot2731\\\\ S_{13}=7167\frac18\quad\qquad\qquad\vee\qquad S_{13}=-2389\frac58[/tex]
