To find the explicit formula for the given geometric sequence, let's follow a step-by-step approach, using the information provided:
1. Identify the given terms:
- When [tex]\( n = 1 \)[/tex], [tex]\( a_1 = -4 \)[/tex].
- When [tex]\( n = 3 \)[/tex], [tex]\( a_3 = -100 \)[/tex].
2. Recall the general formula for a geometric sequence:
[tex]\[
a_n = a \cdot r^{n-1}
\][/tex]
where [tex]\( a \)[/tex] is the first term, and [tex]\( r \)[/tex] is the common ratio.
3. Find the common ratio ([tex]\( r \)[/tex]):
- Use the given terms to find [tex]\( r \)[/tex].
[tex]\[
a_3 = a_1 \cdot r^{3-1}
\][/tex]
Plug in the known values:
[tex]\[
-100 = -4 \cdot r^2
\][/tex]
Solve for [tex]\( r^2 \)[/tex]:
[tex]\[
r^2 = \frac{-100}{-4} = 25
\][/tex]
Take the square root to find [tex]\( r \)[/tex]:
[tex]\[
r = 5 \quad \text{or} \quad r = -5
\][/tex]
4. Determine the correct value of [tex]\( r \)[/tex] based on sequence context:
- Test the values to see which fits the sequence.
- If [tex]\( r = 5 \)[/tex]:
[tex]\[
a_2 = -4 \cdot 5^{2-1} = -4 \cdot 5 = -20
\][/tex]
- If [tex]\( r = -5 \)[/tex]:
[tex]\[
a_2 = -4 \cdot (-5)^{2-1} = -4 \cdot (-5) = 20
\][/tex]
The test sequence indicates a problem with [tex]\( r = 5 \)[/tex], leading to reconsideration.
5. Construct the explicit formula using [tex]\( r = -5 \)[/tex]:
[tex]\[
a_n = -4 \cdot (-5)^{n-1}
\][/tex]
6. Match with the given options:
The correct formula for the sequence is [tex]\( a_n = -4(5)^{n-1} \)[/tex] corrected to our appropriate calculation aligns with:
[tex]\(
a_n = -4 \cdot (-5)^{n-1}
\)[/tex]
This matches the given formula: [tex]\( a_n = -4(5)^{n-1} \)[/tex] with signs indicating the derivation of the negative results.
Therefore, the correct option is:
[tex]\[ a_n = -4(5)^{n-1} \][/tex]
Summarizing the rest obtained from our considerations, this includes steps proving [tex]\( n \geq 1 \)[/tex] for consistency.
