To determine which explicit formula can be used to model the given sequence, we need to test each of the provided formulas and see which one matches the sequence values correctly. Let's take it step by step and substitute [tex]\( x = 1 \)[/tex] into each of the formulas to see the resulting values:
1. [tex]\( f(x) = 12(4)^x \)[/tex]
- When [tex]\( x = 1 \)[/tex]:
[tex]\[
f(1) = 12 \cdot 4^1 = 12 \cdot 4 = 48
\][/tex]
2. [tex]\( f(x) = 3(4)^{x-1} \)[/tex]
- When [tex]\( x = 1 \)[/tex]:
[tex]\[
f(1) = 3 \cdot 4^{1-1} = 3 \cdot 4^0 = 3 \cdot 1 = 3
\][/tex]
3. [tex]\( f(x) = 4(12)^x \)[/tex]
- When [tex]\( x = 1 \)[/tex]:
[tex]\[
f(1) = 4 \cdot 12^1 = 4 \cdot 12 = 48
\][/tex]
4. [tex]\( f(x) = 4(3)^{x-1} \)[/tex]
- When [tex]\( x = 1 \)[/tex]:
[tex]\[
f(1) = 4 \cdot 3^{1-1} = 4 \cdot 3^0 = 4 \cdot 1 = 4
\][/tex]
Based on the calculations:
- For [tex]\( f(x) = 12(4)^x \)[/tex], when [tex]\( x = 1 \)[/tex], the result is 48.
- For [tex]\( f(x) = 3(4)^{x-1} \)[/tex], when [tex]\( x = 1 \)[/tex], the result is 3.
- For [tex]\( f(x) = 4(12)^x \)[/tex], when [tex]\( x = 1 \)[/tex], the result is 48.
- For [tex]\( f(x) = 4(3)^{x-1} \)[/tex], when [tex]\( x = 1 \)[/tex], the result is 4.
Thus, the sequences given by [tex]\( f(x) = 12(4)^x \)[/tex] and [tex]\( f(x) = 4(12)^x \)[/tex] both produce the matching value of 48 when [tex]\( x = 1 \)[/tex].
However, for better pattern matching beyond the initial term:
- [tex]\( f(x) = 12(4)^x \)[/tex] maintains more consistent exponential behavior fitting standard patterns and simplicity.
- [tex]\( f(x) = 4(12)^x \)[/tex] becomes higher value with higher x.
Therefore, [tex]\( f(x) = 12(4)^x \)[/tex] is the explicit formula modeling the same sequence, compatibles and consistent as general fit.
