To determine the first three terms of the geometric sequence given by the expression
[tex]\[ 6 \times 3^{n-1} \][/tex]
we will substitute [tex]\( n \)[/tex] with the values 1, 2, and 3.
1. First Term:
Substitute [tex]\( n = 1 \)[/tex] into the expression:
[tex]\[ 6 \times 3^{1-1} \][/tex]
Since [tex]\( 1-1\)[/tex] equals 0 , we have:
[tex]\[ 6 \times 3^0 \][/tex]
We know that any number raised to the power of 0 is 1:
[tex]\[ 3^0 = 1 \][/tex]
Thus, the first term is:
[tex]\[ 6 \times 1 = 6 \][/tex]
2. Second Term:
Substitute [tex]\( n = 2 \)[/tex] into the expression:
[tex]\[ 6 \times 3^{2-1} \][/tex]
Since [tex]\( 2-1\)[/tex] equals 1 , we have:
[tex]\[ 6 \times 3^1 \][/tex]
We know that any number raised to the power of 1 remains the same:
[tex]\[ 3^1 = 3 \][/tex]
Thus, the second term is:
[tex]\[ 6 \times 3 = 18 \][/tex]
3. Third Term:
Substitute [tex]\( n = 3 \)[/tex] into the expression:
[tex]\[ 6 \times 3^{3-1} \][/tex]
Since [tex]\( 3-1\)[/tex] equals 2 , we have:
[tex]\[ 6 \times 3^2 \][/tex]
We know that [tex]\( 3^2 = 3 \times 3 = 9 \)[/tex]:
[tex]\[ 3^2 = 9 \][/tex]
Thus, the third term is:
[tex]\[ 6 \times 9 = 54 \][/tex]
Therefore, the first three terms of the sequence are:
[tex]\[ 6, 18, 54 \][/tex]
