Certainly! To find the explicit formula for the given geometric sequence: 224, 112, 56, 28, ..., we need to follow these steps:
1. Identify the first term (a₁): The first term of the sequence is given directly.
- So, [tex]\( a_1 = 224 \)[/tex].
2. Determine the common ratio (r): The common ratio can be found by dividing the second term by the first term.
- Common ratio [tex]\( r = \frac{112}{224} = \frac{1}{2} \)[/tex].
3. Formulate the explicit formula:
- The general explicit formula for a geometric sequence is given by:
[tex]\[
a(n) = a_1 \cdot r^{(n-1)}
\][/tex]
- Here, [tex]\( a_1 = 224 \)[/tex] and [tex]\( r = \frac{1}{2} \)[/tex].
4. Substitute the values into the formula:
- Substitute [tex]\( a_1 \)[/tex] and [tex]\( r \)[/tex] into the general formula:
[tex]\[
a(n) = 224 \cdot \left(\frac{1}{2}\right)^{(n-1)}
\][/tex]
Therefore, the explicit formula for the given geometric sequence is:
[tex]\[ a(n) = 224 \cdot \left(\frac{1}{2}\right)^{(n-1)} \][/tex]
Among the given options, this corresponds to:
[tex]\[ a(n) = 224 \left(\frac{1}{2}\right)^{x} \][/tex]
Here, [tex]\( x = n-1 \)[/tex].
Hence, the correct choice is:
[tex]\[ a(n) = 224 \left(\frac{1}{2}\right)^{x} \][/tex]
