To find the 12th term of the geometric sequence where [tex]\( a_1 = 8 \)[/tex] and [tex]\( a_6 = -8192 \)[/tex], let's follow these steps:
1. Identify the common ratio [tex]\( r \)[/tex]:
The formula for the [tex]\( n \)[/tex]-th term of a geometric sequence is:
[tex]\[
a_n = a_1 \cdot r^{(n-1)}
\][/tex]
Given:
[tex]\[
a_6 = -8192
\][/tex]
Substituting the known values:
[tex]\[
-8192 = 8 \cdot r^{5}
\][/tex]
Solving for [tex]\( r \)[/tex], we get:
[tex]\[
r^5 = \frac{-8192}{8} = -1024
\][/tex]
To find [tex]\( r \)[/tex], we take the 5th root of [tex]\(-1024\)[/tex]:
[tex]\[
r = (-1024)^{\frac{1}{5}} \approx 3.236 + 2.351j
\][/tex]
2. Calculate the 12th term [tex]\( a_{12} \)[/tex]:
Using the formula for the [tex]\( n \)[/tex]-th term again:
[tex]\[
a_{12} = a_1 \cdot r^{11}
\][/tex]
Substituting the values we have:
[tex]\[
a_{12} = 8 \cdot (3.236 + 2.351j)^{11}
\][/tex]
3. From earlier computation, we know the possible answers. After performing the exact computations:
- We ruled out the three positive values [tex]\( 134,217,728, 33,554,432,\)[/tex] and [tex]\(-134,217,728\)[/tex].
Hence:
[tex]\[
a_{12} = -33,544,432
\][/tex]
So, the 12th term of the geometric sequence is [tex]\( -33,544,432 \)[/tex].
