Certainly! Let's break down the solution to find the value of [tex]\(a_n\)[/tex] for [tex]\(n=8\)[/tex].
### Step-by-Step Solution:
1. Substitute [tex]\(n = 8\)[/tex] into the sequence formula:
The given sequence formula is:
[tex]\[
a_n = \left( \frac{n}{9} - 12 \right)^n
\][/tex]
Substituting [tex]\(n = 8\)[/tex]:
[tex]\[
a_8 = \left( \frac{8}{9} - 12 \right)^8
\][/tex]
2. Simplify the expression inside the parentheses:
First, we need to calculate [tex]\(\frac{8}{9} - 12\)[/tex]. Let's split this calculation into parts:
- Calculate [tex]\(\frac{8}{9}\)[/tex]:
[tex]\[
\frac{8}{9} \approx 0.8889
\][/tex]
- Subtract 12 from [tex]\(\frac{8}{9}\)[/tex]:
[tex]\[
0.8889 - 12 = -11.1111
\][/tex]
So, the expression inside the parentheses simplifies to:
[tex]\[
-11.1111
\][/tex]
3. Raise the simplified expression to the power of 8:
Now we need to raise [tex]\(-11.1111\)[/tex] to the power of 8:
[tex]\[
a_8 = \left( -11.1111 \right)^8
\][/tex]
4. Calculate the value of [tex]\(\left( -11.1111 \right)^8\)[/tex]:
Finally, we perform the calculation:
[tex]\[
\left( -11.1111 \right)^8 \approx 232305731.25418767
\][/tex]
Therefore, the value of [tex]\(a_8\)[/tex] is:
[tex]\[
\boxed{232305731.25418767}
\][/tex]
