Question

Write the function that represents the sequence in the table.
Start 2 By 5 Table 1st Row 1st Column x 2nd Column 1 3rd Column 2 4st Column 3 5st Column 4 2nd Row 1st Column f left parenthesis x right parenthesis 2nd Column 6 3rd Column 24 4st Column 96 5st Column 384 EndTable

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Part 1
The function that represents the sequence in the table is f left parenthesis x right parenthesis equals
  
enter your response here.
​(Type your answer using exponential​ notation.)

Answer 1

To represent the sequence in the table you've provided, we can analyze the values in the second row:

- For \( x = 1 \), \( f(1) = 6 \)
- For \( x = 2 \), \( f(2) = 24 \)
- For \( x = 3 \), \( f(3) = 96 \)
- For \( x = 4 \), \( f(4) = 384 \)

Let's examine the pattern in the function values:

1. **Find the ratio of consecutive values**:
- \( \frac{f(2)}{f(1)} = \frac{24}{6} = 4 \)
- \( \frac{f(3)}{f(2)} = \frac{96}{24} = 4 \)
- \( \frac{f(4)}{f(3)} = \frac{384}{96} = 4 \)

The function values are multiplied by 4 as \( x \) increases by 1, suggesting an exponential relationship.

2. **Check if the function fits a standard form**:
Given the common ratio, we can hypothesize that \( f(x) = k \cdot 4^{x-1} \) for some constant \( k \).

3. **Determine \( k \)**:
- We can find \( k \) using the value of \( f(1) \):
- \( f(1) = k \cdot 4^{1-1} = k \cdot 1 = k \)
- Since \( f(1) = 6 \), we find that \( k = 6 \).

Thus, we have the function:

\[
f(x) = 6 \cdot 4^{x-1}
\]

This matches the values given in the table. Therefore, the function that represents the sequence in the table is:

\[
\boxed{6 \cdot 4^{x-1}}
\]