To find the first six terms of the sequence defined by [tex]\( a_n = \frac{1}{2}n - 3 \)[/tex], we follow these steps:
1. Determine the first term [tex]\( a_1 \)[/tex] by plugging [tex]\( n = 1 \)[/tex] into the formula:
[tex]\[
a_1 = \frac{1}{2}(1) - 3 = \frac{1}{2} - 3 = -2.5
\][/tex]
So, the first term is [tex]\( -2.5 \)[/tex].
2. Determine the second term [tex]\( a_2 \)[/tex] by plugging [tex]\( n = 2 \)[/tex] into the formula:
[tex]\[
a_2 = \frac{1}{2}(2) - 3 = 1 - 3 = -2.0
\][/tex]
So, the second term is [tex]\( -2.0 \)[/tex].
3. Determine the third term [tex]\( a_3 \)[/tex] by plugging [tex]\( n = 3 \)[/tex] into the formula:
[tex]\[
a_3 = \frac{1}{2}(3) - 3 = 1.5 - 3 = -1.5
\][/tex]
So, the third term is [tex]\( -1.5 \)[/tex].
4. Determine the fourth term [tex]\( a_4 \)[/tex] by plugging [tex]\( n = 4 \)[/tex] into the formula:
[tex]\[
a_4 = \frac{1}{2}(4) - 3 = 2 - 3 = -1.0
\][/tex]
So, the fourth term is [tex]\( -1.0 \)[/tex].
5. Determine the fifth term [tex]\( a_5 \)[/tex] by plugging [tex]\( n = 5 \)[/tex] into the formula:
[tex]\[
a_5 = \frac{1}{2}(5) - 3 = 2.5 - 3 = -0.5
\][/tex]
So, the fifth term is [tex]\( -0.5 \)[/tex].
6. Determine the sixth term [tex]\( a_6 \)[/tex] by plugging [tex]\( n = 6 \)[/tex] into the formula:
[tex]\[
a_6 = \frac{1}{2}(6) - 3 = 3 - 3 = 0.0
\][/tex]
So, the sixth term is [tex]\( 0.0 \)[/tex].
Hence, the first six terms of the sequence [tex]\( a_n = \frac{1}{2} n - 3 \)[/tex] are:
[tex]\[ -2.5, -2.0, -1.5, -1.0, -0.5, 0.0 \][/tex]
