To find the first five terms of the geometric sequence with the first term [tex]\( a_1 = 4 \)[/tex] and the common ratio [tex]\( r = 3 \)[/tex], we use the formula for the [tex]\( n \)[/tex]-th term of a geometric sequence:
[tex]\[ a_n = a_1 \cdot r^{n-1} \][/tex]
Let's calculate each term step-by-step:
1. First term ([tex]\(a_1\)[/tex]):
[tex]\[ a_1 = 4 \][/tex]
2. Second term ([tex]\(a_2\)[/tex]):
[tex]\[ a_2 = a_1 \cdot r = 4 \cdot 3 = 12 \][/tex]
3. Third term ([tex]\(a_3\)[/tex]):
[tex]\[ a_3 = a_1 \cdot r^2 = 4 \cdot (3^2) = 4 \cdot 9 = 36 \][/tex]
4. Fourth term ([tex]\(a_4\)[/tex]):
[tex]\[ a_4 = a_1 \cdot r^3 = 4 \cdot (3^3) = 4 \cdot 27 = 108 \][/tex]
5. Fifth term ([tex]\(a_5\)[/tex]):
[tex]\[ a_5 = a_1 \cdot r^4 = 4 \cdot (3^4) = 4 \cdot 81 = 324 \][/tex]
Therefore, the first five terms of the geometric sequence are:
[tex]\[ 4, 12, 36, 108, 324 \][/tex]
So, the first term is [tex]\(\boxed{4}\)[/tex].
