To find the [tex]$11^{\text{th}}$[/tex] term of the given sequence [tex]\(8, 24, 72, 216, \ldots\)[/tex], let's first understand the pattern of the sequence. This sequence is a geometric sequence.
In a geometric sequence, each term is obtained by multiplying the previous term with a constant ratio. Let's identify this ratio:
1. The first term is [tex]\(8\)[/tex].
2. To obtain the second term, we multiply the first term by the ratio [tex]\(r\)[/tex]:
[tex]\[
8 \times r = 24
\][/tex]
Solving for [tex]\(r\)[/tex]:
[tex]\[
r = \frac{24}{8} = 3
\][/tex]
3. To confirm, we check this ratio for the next term:
[tex]\[
24 \times 3 = 72
\][/tex]
[tex]\[
72 \times 3 = 216
\][/tex]
So, the common ratio [tex]\(r\)[/tex] is [tex]\(3\)[/tex].
The general formula for the [tex]\(n\)[/tex]-th term of a geometric sequence is:
[tex]\[
a_n = a \times r^{n-1}
\][/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:
- [tex]\(a = 8\)[/tex]
- [tex]\(r = 3\)[/tex]
- [tex]\(n = 11\)[/tex]
We need to find the [tex]\(11^{\text{th}}\)[/tex] term ([tex]\(a_{11}\)[/tex]):
[tex]\[
a_{11} = 8 \times 3^{11-1}
\][/tex]
[tex]\[
a_{11} = 8 \times 3^{10}
\][/tex]
Thus, the [tex]$11^{\text{th}}$[/tex] term of the sequence is:
[tex]\[
472392
\][/tex]
The [tex]$11^{\text{th}}$[/tex] term of the sequence is [tex]\( \boxed{472392} \)[/tex].
