To find the first five terms of the sequence given by the formula [tex]\( a_n = 6n^2 - 4n \)[/tex], we will substitute [tex]\( n \)[/tex] with 1, 2, 3, 4, and 5 respectively.
1. Calculate [tex]\( a_1 \)[/tex]:
[tex]\[
a_1 = 6 \cdot 1^2 - 4 \cdot 1
\][/tex]
[tex]\[
a_1 = 6 \cdot 1 - 4
\][/tex]
[tex]\[
a_1 = 6 - 4
\][/tex]
[tex]\[
a_1 = 2
\][/tex]
2. Calculate [tex]\( a_2 \)[/tex]:
[tex]\[
a_2 = 6 \cdot 2^2 - 4 \cdot 2
\][/tex]
[tex]\[
a_2 = 6 \cdot 4 - 4 \cdot 2
\][/tex]
[tex]\[
a_2 = 24 - 8
\][/tex]
[tex]\[
a_2 = 16
\][/tex]
3. Calculate [tex]\( a_3 \)[/tex]:
[tex]\[
a_3 = 6 \cdot 3^2 - 4 \cdot 3
\][/tex]
[tex]\[
a_3 = 6 \cdot 9 - 4 \cdot 3
\][/tex]
[tex]\[
a_3 = 54 - 12
\][/tex]
[tex]\[
a_3 = 42
\][/tex]
4. Calculate [tex]\( a_4 \)[/tex]:
[tex]\[
a_4 = 6 \cdot 4^2 - 4 \cdot 4
\][/tex]
[tex]\[
a_4 = 6 \cdot 16 - 4 \cdot 4
\][/tex]
[tex]\[
a_4 = 96 - 16
\][/tex]
[tex]\[
a_4 = 80
\][/tex]
5. Calculate [tex]\( a_5 \)[/tex]:
[tex]\[
a_5 = 6 \cdot 5^2 - 4 \cdot 5
\][/tex]
[tex]\[
a_5 = 6 \cdot 25 - 4 \cdot 5
\][/tex]
[tex]\[
a_5 = 150 - 20
\][/tex]
[tex]\[
a_5 = 130
\][/tex]
Therefore, the first five terms of the sequence are:
[tex]\[
a_1 = 2, \quad a_2 = 16, \quad a_3 = 42, \quad a_4 = 80, \quad a_5 = 130.
\][/tex]
