Answer
$${a_1} = 1,\,\,\,\,d = - \frac{{20}}{{11}}$$
Work Step by Step
$$\eqalign{
& {S_{12}} = - 108,\,\,\,{a_{12}} = - 19 \cr
& {\text{Use the first formula for }}{S_n} \cr
& {S_n} = \frac{n}{2}\left( {{a_1} + {a_n}} \right) \cr
& {\text{Let }}n = 12 \cr
& {S_{12}} = \frac{{12}}{2}\left( {{a_1} + {a_{12}}} \right) \cr
& - 108 = 6\left( {{a_1} - 19} \right) \cr
& {\text{Solve for }}{a_1} \cr
& - 18 = {a_1} - 19 \cr
& {a_1} = 1 \cr
& \cr
& {\text{Now find }}d,{\text{ use the formula for the }}n{\text{th term}} \cr
& {a_n} = {a_1} + \left( {n - 1} \right)d \cr
& {a_{12}} = {a_1} + \left( {12 - 1} \right)d \cr
& - 19 = 1 + \left( {11} \right)d \cr
& - 20 = 11d \cr
& d = - \frac{{20}}{{11}} \cr
& \cr
& {\text{Then,}} \cr
& {a_1} = 1,\,\,\,\,d = - \frac{{20}}{{11}} \cr} $$
