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 30 4st Column 150 5st Column 750 EndTable

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Part 1
The function that represents the sequence in the table is f left parenthesis x right parenthesis equals
  
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Answer 1

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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