To determine the common ratio of the associated geometric sequence from the table, let's analyze the [tex]\( y \)[/tex] values:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
1 & 7 \\
\hline
2 & 21 \\
\hline
3 & 63 \\
\hline
4 & 189 \\
\hline
5 & 567 \\
\hline
\end{array}
\][/tex]
The common ratio [tex]\( r \)[/tex] of a geometric sequence can be found by dividing any term in the sequence by the preceding term. Let's calculate the common ratio step-by-step for the given [tex]\( y \)[/tex] values:
1. Calculate the ratio between the second term and the first term:
[tex]\[
r = \frac{y_2}{y_1} = \frac{21}{7} = 3
\][/tex]
2. Calculate the ratio between the third term and the second term:
[tex]\[
r = \frac{y_3}{y_2} = \frac{63}{21} = 3
\][/tex]
3. Calculate the ratio between the fourth term and the third term:
[tex]\[
r = \frac{y_4}{y_3} = \frac{189}{63} = 3
\][/tex]
4. Calculate the ratio between the fifth term and the fourth term:
[tex]\[
r = \frac{y_5}{y_4} = \frac{567}{189} = 3
\][/tex]
Since the ratio between each pair of consecutive terms is consistently 3, the common ratio for this geometric sequence is [tex]\( 3 \)[/tex].
Thus, the correct answer is:
D. 3
