To determine the common ratio of the associated geometric sequence, follow these steps:
1. Understand the Sequence: The values given are [tex]\( y \)[/tex]-values of an exponential function corresponding to integer [tex]\( x \)[/tex]-values. The [tex]\( y \)[/tex]-values are [tex]\( 6, 36, 216, 1296, \)[/tex] and [tex]\( 7776 \)[/tex].
2. Identify the Formula: In a geometric sequence, each term after the first is found by multiplying the previous term by a constant common ratio, denoted as [tex]\( r \)[/tex].
3. Determine the Common Ratio: To find the common ratio [tex]\( r \)[/tex], divide any term by the preceding term in the sequence.
4. Calculate [tex]\( r \)[/tex]:
- Take the second term [tex]\( y_2 = 36 \)[/tex] and divide it by the first term [tex]\( y_1 = 6 \)[/tex]:
[tex]\[
r = \frac{y_2}{y_1} = \frac{36}{6} = 6
\][/tex]
So, the common ratio is [tex]\( 6 \)[/tex].
5. Verify Consistency: Ensure this ratio applies consistently throughout the sequence:
[tex]\[
r = \frac{y_3}{y_2} = \frac{216}{36} = 6
\][/tex]
[tex]\[
r = \frac{y_4}{y_3} = \frac{1296}{216} = 6
\][/tex]
[tex]\[
r = \frac{y_5}{y_4} = \frac{7776}{1296} = 6
\][/tex]
Since the common ratio [tex]\( r \)[/tex] is consistently [tex]\( 6 \)[/tex] throughout the sequence, the common ratio of the associated geometric sequence is [tex]\( 6 \)[/tex].
Thus, the answer is [tex]\( B. 6 \)[/tex].
