To find the third term of a geometric sequence given the first term ([tex]\(a_1\)[/tex]) and the common ratio ([tex]\(r\)[/tex]), we use the general formula for the nth term of a geometric sequence:
[tex]\[
a_n = a_1 \cdot r^{n-1}
\][/tex]
In this case, the first term [tex]\(a_1\)[/tex] is [tex]\(\frac{1}{4}\)[/tex] and the common ratio [tex]\(r\)[/tex] is [tex]\(-2\)[/tex]. We need to find the third term ([tex]\(a_3\)[/tex]) of the sequence. Plugging in the values:
[tex]\[
a_3 = a_1 \cdot r^{3-1} = a_1 \cdot r^2
\][/tex]
Substituting [tex]\(a_1\)[/tex] and [tex]\(r\)[/tex]:
[tex]\[
a_3 = \frac{1}{4} \cdot (-2)^2
\][/tex]
Next, calculate [tex]\((-2)^2\)[/tex]:
[tex]\[
(-2)^2 = (-2) \cdot (-2) = 4
\][/tex]
So, the expression simplifies to:
[tex]\[
a_3 = \frac{1}{4} \cdot 4
\][/tex]
Multiplying the two values:
[tex]\[
a_3 = 1
\][/tex]
Therefore, the third term of the geometric sequence is:
[tex]\[
1.0
\][/tex]
