https://file.aiquickdraw.com/mathbot/file/1725468672606s6gfoum3.jpg

The image you've provided shows a table with values of \( n \) and \( f(n) \). The table looks like this:

\[
\begin{array}{c|c}
n & f(n) \\
\hline
0 & 27 \\
1 & 9 \\
2 & 3 \\
3 & 1 \\
4 & ? \\
5 & ? \\
6 & ? \\
\end{array}
\]

Based on the values given for \( n = 0, 1, 2, 3 \), the function \( f(n) \) appears to decrease by a factor of 3 each time \( n \) increases by 1. This suggests that \( f(n) \) could be a geometric sequence, where the common ratio \( r = \frac{1}{3} \).

If this pattern continues, then the formula for \( f(n) \) can be expressed as:

\[
f(n) = 27 \times \left(\frac{1}{3}\right)^n
\]

Let's verify and find the missing values for \( n = 4, 5, 6 \):

- \( f(4) = 27 \times \left(\frac{1}{3}\right)^4 = 27 \times \frac{1}{81} = \frac{27}{81} = \frac{1}{3} \)
- \( f(5) = 27 \times \left(\frac{1}{3}\right)^5 = 27 \times \frac{1}{243} = \frac{27}{243} = \frac{1}{9} \)
- \( f(6) = 27 \times \left(\frac{1}{3}\right)^6 = 27 \times \frac{1}{729} = \frac{27}{729} = \frac{1}{27} \)

Thus, the completed table should look like this:

\[
\begin{array}{c|c}
n & f(n) \\
\hline
0 & 27 \\
1 & 9 \\
2 & 3 \\
3 & 1 \\
4 & \frac{1}{3} \\
5 & \frac{1}{9} \\
6 & \frac{1}{27} \\
\end{array}
\]

Would you like more details on geometric sequences or have any questions?

### Related Questions:
1. How do you find the sum of a finite geometric series?
2. What is the general formula for a geometric sequence?
3. How do geometric sequences apply in real-world situations?
4. How would the sequence change if the common ratio were negative?
5. What are the properties of geometric sequences?

### Tip:
When identifying sequences, always check the differences (for arithmetic sequences) or ratios (for geometric sequences) between consecutive terms to determine the pattern.

Determine the next three values in the sequence based on the given table where n ranges from 0 to 6.

Learn how to determine the missing values in a geometric sequence where the initial value is 27 and the common ratio is 1/3. This problem is suitable for students in Grades 7-9 and involves key algebraic concepts and geometric sequences.

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