To find the function that represents the sequence given in the table, we can first analyze the values of \( f(x) \) for the corresponding values of \( x \):
The table provides:
- For \( x = 1 \), \( f(1) = 6 \)
- For \( x = 2 \), \( f(2) = 30 \)
- For \( x = 3 \), \( f(3) = 150 \)
- For \( x = 4 \), \( f(4) = 750 \)
Now, let's observe if there is a pattern or formula we can derive from these values.
We can look at the ratios of consecutive \( f(x) \):
- \( \frac{f(2)}{f(1)} = \frac{30}{6} = 5 \)
- \( \frac{f(3)}{f(2)} = \frac{150}{30} = 5 \)
- \( \frac{f(4)}{f(3)} = \frac{750}{150} = 5 \)
The ratio between consecutive terms is constant (each term is multiplying the previous term by 5). Thus, it appears that \( f(x) \) could be expressed as \( f(x) = f(1) \times 5^{(x-1)} \).
Since \( f(1) = 6 \), we can express the function as: \[ f(x) = 6 \times 5^{(x-1)} \]
Thus, the function that represents the sequence in the table is: \[ f(x) = 6 \times 5^{(x-1)} \]
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