To determine which formula describes the sequence [tex]\(-3, \frac{3}{5}, -\frac{3}{25}, \frac{3}{125}, -\frac{3}{625}\)[/tex], let's analyze the pattern and components of the sequence step-by-step.
1. Identify the initial term and the progression:
- The initial term (first term) of the sequence is [tex]\(-3\)[/tex].
2. Determine the common ratio:
- The sequence alternates in sign and the absolute value of each term is multiplied by [tex]\(\frac{1}{5}\)[/tex].
- Thus, the common ratio will involve both changing the sign and multiplying by [tex]\(\frac{1}{5}\)[/tex].
3. Analyze the sign pattern:
- The sequence alternates in sign: negative, positive, negative, positive, negative, and so on.
- This suggests that the formula involves [tex]\((-1)^{x-1}\)[/tex] to alternate the signs.
4. Combine initial term, ratio, and signs:
- The general term of the sequence can therefore be written incorporating all these factors:
[tex]\[
f(x) = -3 \cdot \left( -\frac{1}{5} \right)^{x-1}
\][/tex]
- Here [tex]\(-3\)[/tex] is the initial term, [tex]\(-\frac{1}{5}\)[/tex] accounts for both the negative sign and the denominator of each successive term powered to [tex]\((x-1)\)[/tex].
Putting it all together, the formula that correctly describes the sequence is:
[tex]\[
f(x) = -3 \left( -\frac{1}{5} \right)^{x-1}
\][/tex]
Thus, the correct formula for the given sequence is:
[tex]\[
f(x) = -3\left( -\frac{1}{5} \right)^{x-1}
\][/tex]
