Answer
$${\text{ }}{a_1} = 1,{\text{ }}{a_2} = 0,{\text{ }}{a_3} = - 1,{\text{ }}{a_4} = 0$$
Work Step by Step
$$\eqalign{
& {a_{n + 1}} = a_n^2 - 1;{\text{ }}{a_1} = 1 \cr
& \cr
& {\text{Let }}n = 1 \cr
& {a_{1 + 1}} = a_1^2 - 1 \cr
& {a_2} = a_1^2 - 1 \cr
& {\text{Substitute }}{a_1}{\text{ and simplify}} \cr
& {a_2} = {\left( 1 \right)^2} - 1 = 0 \cr
& \cr
& {\text{Let }}n = 2 \cr
& {a_{2 + 1}} = a_2^2 - 1 \cr
& {a_3} = a_2^2 - 1 \cr
& {\text{Substitute }}{a_2}{\text{ and simplify}} \cr
& {a_3} = {\left( 0 \right)^2} - 1 = - 1 \cr
& \cr
& {\text{Let }}n = 3 \cr
& {a_{3 + 1}} = a_3^2 - 1 \cr
& {a_4} = a_3^2 - 1 \cr
& {\text{Substitute }}{a_2}{\text{ and simplify}} \cr
& {a_4} = {\left( { - 1} \right)^2} - 1 = 0 \cr
& \cr
& {\text{Therefore, the first four terms of the sequence are:}} \cr
& {\text{ }}{a_1} = 1,{\text{ }}{a_2} = 0,{\text{ }}{a_3} = - 1,{\text{ }}{a_4} = 0 \cr} $$
