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The values in the table represent an exponential function. What is the common ratio of the associated geometric sequence?

\begin{tabular}{|c|c|}
\hline
[tex]$x$[/tex] & [tex]$y$[/tex] \\
\hline
1 & 7 \\
\hline
2 & 21 \\
\hline
3 & 63 \\
\hline
4 & 189 \\
\hline
5 & 567 \\
\hline
\end{tabular}

A. 28
B. 7
C. 14
D. 3

Sagot :

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