[tex]\displaystyle\bf\\a_{1} =-8\\\\q=3\\\\S_{n} =-2912\\\\n=?\\\\\\S_{n} =\frac{a_{1} \cdot(q^{n} -1)}{q-1} \\\\\\-2912=\frac{-8\cdot(3^{n} -1)}{3-1} \\\\\\-2912=-4\cdot(3^{n} -1)\\\\\\3^{n} -1=-2912:(-4)\\\\\\3^{n} -1=728\\\\\\3^{n} =729\\\\\\3^{n } =3^{6} \\\\\\\boxed{n=6}[/tex]