To predict the general term, or nth term, [tex]\( a_n \)[/tex], of the given sequence [tex]\(-12, 36, -108, 324, -972, \ldots\)[/tex], we need to follow these steps:
1. Identify the type of sequence: The sequence appears to be a geometric sequence because each term is obtained by multiplying the previous term by a constant ratio.
2. Determine the common ratio: To find the common ratio [tex]\( r \)[/tex], divide any term by the preceding term.
[tex]\[
r = \frac{36}{-12} = -3
\][/tex]
[tex]\[
r = \frac{-108}{36} = -3
\][/tex]
[tex]\[
r = \frac{324}{-108} = -3
\][/tex]
This confirms that the common ratio [tex]\( r \)[/tex] is [tex]\(-3\)[/tex].
3. Identify the first term: The first term [tex]\( a \)[/tex] of the sequence is [tex]\(-12\)[/tex].
4. Write the general term of the geometric sequence: The general formula for the nth term of a geometric sequence is given by:
[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.
5. Substitute the values of [tex]\( a \)[/tex] and [tex]\( r \)[/tex]: Here, [tex]\( a = -12 \)[/tex] and [tex]\( r = -3 \)[/tex]. Substituting these values into the formula gives:
[tex]\[
a_n = -12 \cdot (-3)^{n-1}
\][/tex]
Therefore, the general term [tex]\( a_n \)[/tex] of the sequence [tex]\(-12, 36, -108, 324, -972, \ldots\)[/tex] is:
[tex]\[
a_n = -12 \cdot (-3)^{n-1}
\][/tex]
