To derive an explicit formula for the sequence \( a_n \), where the terms are \( 8, 4, 2, \ldots \), we can observe the following:
- The first term \( a_1 = 8 \).
- The second term \( a_2 = 4 \).
- The third term \( a_3 = 2 \).
Each term is getting halved compared to the previous term. Thus, we can express each term in terms of the previous one:
\[ a_n = \frac{a_{n-1}}{2} \]
This indicates a geometric sequence, where the first term is \( a_1 = 8 \) and the common ratio \( r = \frac{1}{2} \).
The general formula for the \( n \)-th term of a geometric sequence is given by:
\[ a_n = a_1 \cdot r^{n-1} \]
Plugging in the values we have:
- \( a_1 = 8 \)
- \( r = \frac{1}{2} \)
So we obtain:
\[ a_n = 8 \cdot \left( \frac{1}{2} \right)^{n-1} \]
This can also be rewritten as:
\[ a_n = \frac{8}{2^{n-1}} \]
Thus, the explicit formula for the \( n \)-th term of the sequence is:
\[ a_n = \frac{8}{2^{n-1}} \]