To find the general term, [tex]\( a_n \)[/tex], of the given sequence:
[tex]\[
-12, 48, -192, 768, -3072, \ldots
\][/tex]
we proceed as follows:
1. Identify the first term [tex]\( a_1 \)[/tex]:
The first term of the sequence is [tex]\( a_1 = -12 \)[/tex].
2. Determine the common ratio [tex]\( r \)[/tex]:
To find the common ratio, we divide the second term by the first term:
[tex]\[
r = \frac{48}{-12} = -4
\][/tex]
We can double-check this by considering the third and second terms:
[tex]\[
r = \frac{-192}{48} = -4
\][/tex]
3. Form the general term:
In a geometric sequence, the general term [tex]\( a_n \)[/tex] can be expressed as:
[tex]\[
a_n = a_1 \cdot r^{n-1}
\][/tex]
Substituting the values of [tex]\( a_1 \)[/tex] and [tex]\( r \)[/tex]:
[tex]\[
a_n = -12 \cdot (-4)^{n-1}
\][/tex]
Thus, the general term [tex]\( a_n \)[/tex] of the sequence is:
[tex]\[
\boxed{a_n = -12 \cdot (-4)^{n-1}}
\][/tex]
