To determine the type of sequence represented by the given table, we need to examine the relationship between the successive [tex]\( y \)[/tex]-values.
Let's begin by listing the [tex]\( y \)[/tex]-values from the table:
[tex]\[ y_1 = 5, \, y_2 = 3, \, y_3 = 1.8, \, y_4 = 1.08 \][/tex]
### Step 1: Check for an Arithmetic Sequence
An arithmetic sequence has a common difference [tex]\( d \)[/tex] between successive terms. To check this, we calculate the differences between consecutive [tex]\( y \)[/tex]-values:
[tex]\[ \text{Difference between } y_2 \text{ and } y_1 = y_2 - y_1 = 3 - 5 = -2 \][/tex]
[tex]\[ \text{Difference between } y_3 \text{ and } y_2 = y_3 - y_2 = 1.8 - 3 = -1.2 \][/tex]
[tex]\[ \text{Difference between } y_4 \text{ and } y_3 = y_4 - y_3 = 1.08 - 1.8 = -0.72 \][/tex]
Since the differences (-2, -1.2, -0.72) are not constant, the sequence is not arithmetic.
### Step 2: Check for a Geometric Sequence
A geometric sequence has a common ratio [tex]\( r \)[/tex] between successive terms. To check this, we calculate the ratios of consecutive [tex]\( y \)[/tex]-values:
[tex]\[ \text{Ratio between } y_2 \text{ and } y_1 = \frac{y_2}{y_1} = \frac{3}{5} = 0.6 \][/tex]
[tex]\[ \text{Ratio between } y_3 \text{ and } y_2 = \frac{y_3}{y_2} = \frac{1.8}{3} = 0.6 \][/tex]
[tex]\[ \text{Ratio between } y_4 \text{ and } y_3 = \frac{y_4}{y_3} = \frac{1.08}{1.8} = 0.6 \][/tex]
Since the ratios (0.6, 0.6, 0.6) are constant, the sequence is a geometric sequence with a common ratio of [tex]\( 0.6 \)[/tex].
### Conclusion
The table clearly represents a geometric sequence because the successive [tex]\( y \)[/tex]-values have a common ratio of [tex]\( 0.6 \)[/tex].
Therefore, the correct answer is:
[tex]\[ \boxed{\text{C. The table represents a geometric sequence because the successive } y\text{-values have a common ratio of } 0.6} \][/tex]
