To find the first five terms of the sequence defined by the conditions [tex]\( t_1 = -1 \)[/tex] and [tex]\( t_n = t_{n-1} + 2 \)[/tex] for [tex]\( n \geq 2 \)[/tex], we will calculate each term step by step based on the given recursive relation.
1. First Term:
- The first term, [tex]\( t_1 \)[/tex], is given directly:
[tex]\[
t_1 = -1
\][/tex]
2. Second Term:
- To find the second term, [tex]\( t_2 \)[/tex], we use the relationship [tex]\( t_n = t_{n-1} + 2 \)[/tex] with [tex]\( n = 2 \)[/tex]:
[tex]\[
t_2 = t_1 + 2 = -1 + 2 = 1
\][/tex]
3. Third Term:
- To find the third term, [tex]\( t_3 \)[/tex], we use the same relationship with [tex]\( n = 3 \)[/tex]:
[tex]\[
t_3 = t_2 + 2 = 1 + 2 = 3
\][/tex]
4. Fourth Term:
- To find the fourth term, [tex]\( t_4 \)[/tex], we use the same relationship with [tex]\( n = 4 \)[/tex]:
[tex]\[
t_4 = t_3 + 2 = 3 + 2 = 5
\][/tex]
5. Fifth Term:
- To find the fifth term, [tex]\( t_5 \)[/tex], we use the same relationship with [tex]\( n = 5 \)[/tex]:
[tex]\[
t_5 = t_4 + 2 = 5 + 2 = 7
\][/tex]
Thus, the first five terms of the sequence are:
[tex]\[
t_1 = -1, \quad t_2 = 1, \quad t_3 = 3, \quad t_4 = 5, \quad t_5 = 7
\][/tex]
Therefore, the first five terms of the sequence are [tex]\(\boxed{-1, 1, 3, 5, 7}\)[/tex].
