To determine the next term in the geometric sequence [tex]\(-1, -3, -9, -27, -81, \ldots\)[/tex], we need to start by identifying the common ratio and then use it to find the next term.
### 1. Identify the Common Ratio
A geometric sequence is defined by a constant ratio between successive terms. We find this ratio (let's call it [tex]\( r \)[/tex]) by dividing any term by the preceding term. For example:
[tex]\[ r = \frac{{\text{{second term}}}}{{\text{{first term}}}} = \frac{{-3}}{{-1}} = 3 \][/tex]
### 2. Verify the Common Ratio
To ensure our common ratio is correct, we can verify it with additional terms:
[tex]\[ r = \frac{{\text{{third term}}}}{{\text{{second term}}}} = \frac{{-9}}{{-3}} = 3 \][/tex]
[tex]\[ r = \frac{{\text{{fourth term}}}}{{\text{{third term}}}} = \frac{{-27}}{{-9}} = 3 \][/tex]
[tex]\[ r = \frac{{\text{{fifth term}}}}{{\text{{fourth term}}}} = \frac{{-81}}{{-27}} = 3 \][/tex]
Since the common ratio is consistently 3 for these terms, we can be confident that [tex]\( r = 3 \)[/tex] is correct.
### 3. Determine the Next Term
The [tex]\( n \)[/tex]th term of a geometric sequence can be found using the formula:
[tex]\[ a_n = a \cdot r^{(n-1)} \][/tex]
where [tex]\( a \)[/tex] is the first term, [tex]\( r \)[/tex] is the common ratio, and [tex]\( n \)[/tex] is the term number.
Given the first term [tex]\( a = -1 \)[/tex] and the common ratio [tex]\( r = 3 \)[/tex]:
To find the 6th term (since we have 5 terms, the 6th term is the next one):
[tex]\[ a_6 = -1 \cdot 3^{(6-1)} = -1 \cdot 3^5 \][/tex]
Calculate [tex]\( 3^5 \)[/tex]:
[tex]\[ 3^5 = 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 = 243 \][/tex]
Then:
[tex]\[ a_6 = -1 \cdot 243 = -243 \][/tex]
### 4. Verify Against Given Choices
Comparing the calculated next term with the given choices:
- [tex]\(-243\)[/tex] is one of the options given.
Thus, the next term in the geometric sequence is:
[tex]\[ \boxed{-243} \][/tex]
