Answer
$${a_1} = - 11,\,\,\,\,{a_n} = 2n - 13$$
Work Step by Step
$$\eqalign{
& {a_5} = - 3{\text{ and }}{a_{15}} = 17 \cr
& {\text{We obtain }}{a_{15}}{\text{ by adding the common difference to }}{a_5}{\text{ ten times}} \cr
& {a_{15}} = {a_5} + 10d \cr
& {\text{Substituting }}{a_5}{\text{ and }}{a_{15}} \cr
& 17 = - 3 + 10d \cr
& {\text{Solve for }}d \cr
& 20 = 10d \cr
& d = 2 \cr
& \cr
& {\text{The }}n{\text{th Term of an Arithmetic Sequence is given by}} \cr
& {a_n} = {a_1} + \left( {n - 1} \right)d \cr
& {\text{For }}n = 15 \cr
& {a_{15}} = {a_1} + \left( {15 - 1} \right)d \cr
& 17 = {a_1} + \left( {15 - 1} \right)\left( 2 \right) \cr
& 17 = {a_1} + 28 \cr
& {a_1} = - 11 \cr
& \cr
& {a_n} = - 11 + \left( {n - 1} \right)\left( 2 \right) \cr
& {a_n} = - 11 + 2n - 2 \cr
& {a_n} = 2n - 13 \cr
& \cr
& {a_1} = - 11,\,\,\,\,{a_n} = 2n - 13 \cr} $$
