To determine the type of sequence represented in the table, let's analyze the values of [tex]\( y \)[/tex] corresponding to the given values of [tex]\( x \)[/tex]:
[tex]\[
\begin{array}{|c|c|c|c|c|}
\hline
x & 1 & 2 & 3 & 4 \\
\hline
y & 4 & -9.6 & 23.04 & -55.296 \\
\hline
\end{array}
\][/tex]
We need to check if these values form a geometric sequence by examining the common ratio between successive [tex]\( y \)[/tex]-values. In a geometric sequence, the ratio of [tex]\( y_{n+1} \)[/tex] to [tex]\( y_n \)[/tex] (where [tex]\( y_n \)[/tex] is the n-th term) should be constant.
Firstly, let's calculate the ratios between successive terms:
1. Ratio between the second term and the first term:
[tex]\[
\frac{y_2}{y_1} = \frac{-9.6}{4} = -2.4
\][/tex]
2. Ratio between the third term and the second term:
[tex]\[
\frac{y_3}{y_2} = \frac{23.04}{-9.6} = -2.4
\][/tex]
3. Ratio between the fourth term and the third term:
[tex]\[
\frac{y_4}{y_3} = \frac{-55.296}{23.04} = -2.4
\][/tex]
Since the ratio between successive [tex]\( y \)[/tex]-values is consistently [tex]\(-2.4\)[/tex], we can confirm that the given sequence is a geometric sequence with a common ratio of [tex]\(-2.4\)[/tex].
Thus, the correct answer is:
A. The table represents a geometric sequence because the successive [tex]\( y \)[/tex]-values have a common ratio of [tex]\(-2.4\)[/tex].
