To find the third term in the geometric sequence [tex]\( f(n) = 2 \cdot (0.8)^{n-1} \)[/tex], follow these steps:
1. Identify the general form of the geometric sequence:
[tex]\[
f(n) = a \cdot r^{n-1}
\][/tex]
where [tex]\( a \)[/tex] is the first term, [tex]\( r \)[/tex] is the common ratio, and [tex]\( n \)[/tex] is the term number.
2. Given the sequence, [tex]\( a = 2 \)[/tex] and [tex]\( r = 0.8 \)[/tex].
3. We are asked to find the third term ([tex]\( n = 3 \)[/tex]). Substitute [tex]\( n = 3 \)[/tex] into the general form of the sequence:
[tex]\[
f(3) = 2 \cdot (0.8)^{3-1}
\][/tex]
4. Simplify the exponent [tex]\( 3-1 \)[/tex]:
[tex]\[
f(3) = 2 \cdot (0.8)^2
\][/tex]
5. Calculate [tex]\( (0.8)^2 \)[/tex]:
[tex]\[
(0.8)^2 = 0.64
\][/tex]
6. Multiply [tex]\( 2 \)[/tex] by [tex]\( 0.64 \)[/tex]:
[tex]\[
f(3) = 2 \cdot 0.64 = 1.28
\][/tex]
The third term in the geometric sequence [tex]\( f(n) = 2 \cdot (0.8)^{n-1} \)[/tex] is [tex]\( 1.28 \)[/tex].
Therefore, the answer is [tex]\(\boxed{1.28}\)[/tex].
