To determine the type of sequence represented by the given table, we need to analyze the [tex]$y$[/tex]-values at each step and decide if they show a common difference (arithmetic sequence) or a common ratio (geometric sequence).
Given table:
\begin{tabular}{|c|c|c|c|c|}
\hline
[tex]$x$[/tex] & [tex]$1$[/tex] & [tex]$2$[/tex] & [tex]$3$[/tex] & [tex]$4$[/tex] \\
\hline
[tex]$y$[/tex] & [tex]$4$[/tex] & [tex]$-9.6$[/tex] & [tex]$23.04$[/tex] & [tex]$-55.296$[/tex] \\
\hline
\end{tabular}
1. Check for Arithmetic Sequence:
An arithmetic sequence has a common difference ([tex]$d$[/tex]) between successive terms, which means:
[tex]\[
y_{i+1} - y_i = d
\][/tex]
- Calculate the difference between the first and second [tex]$y$[/tex]-values: [tex]\(-9.6 - 4 = -13.6\)[/tex]
- Calculate the difference between the second and third [tex]$y$[/tex]-values: [tex]\(23.04 - (-9.6) = 23.04 + 9.6 = 32.64\)[/tex]
- Calculate the difference between the third and fourth [tex]$y$[/tex]-values: [tex]\(-55.296 - 23.04 = -55.296 - 23.04 = -78.336\)[/tex]
Since the differences are not the same, the given table does not represent an arithmetic sequence.
2. Check for Geometric Sequence:
A geometric sequence has a common ratio ([tex]$r$[/tex]) between successive terms, which means:
[tex]\[
\frac{y_{i+1}}{y_i} = r
\][/tex]
- Calculate the ratio between the first and second [tex]$y$[/tex]-values: [tex]\(\frac{-9.6}{4} = -2.4\)[/tex]
- Calculate the ratio between the second and third [tex]$y$[/tex]-values: [tex]\(\frac{23.04}{-9.6} = -2.4\)[/tex]
- Calculate the ratio between the third and fourth [tex]$y$[/tex]-values: [tex]\(\frac{-55.296}{23.04} = -2.4\)[/tex]
Since all the ratios are equal to [tex]\(-2.4\)[/tex], the table represents a geometric sequence with a common ratio of [tex]\(-2.4\)[/tex].
Therefore, the correct answer is:
B. The table represents a geometric sequence because the successive [tex]$y$[/tex]-values have a common ratio of -2.4.
