To write the explicit rule represented by the given geometric sequence and identify which statements are true, let's analyze the given sequence step-by-step.
We have the sequence:
[tex]\[ \{8, 72, 648, 5832, 52488\} \][/tex]
1. Identifying the Common Ratio:
The common ratio, [tex]\( r \)[/tex], between consecutive terms in a geometric sequence is found by dividing any term by the previous term.
Let's find the common ratio using the first two terms:
[tex]\[ r = \frac{72}{8} = 9 \][/tex]
Since this is a geometric sequence, the same ratio should apply to the other terms as well:
[tex]\[ r = \frac{648}{72} = 9 \][/tex]
[tex]\[ r = \frac{5832}{648} = 9 \][/tex]
[tex]\[ r = \frac{52488}{5832} = 9 \][/tex]
Therefore, the common ratio [tex]\( r \)[/tex] is [tex]\( 9 \)[/tex].
2. Writing the Explicit Rule:
In a geometric sequence, an explicit rule can generally be written in the form:
[tex]\[ f(n) = a \cdot r^{(n-1)} \][/tex]
where [tex]\( a \)[/tex] is the first term and [tex]\( r \)[/tex] is the common ratio.
Given:
[tex]\[ a = 8 \][/tex]
[tex]\[ r = 9 \][/tex]
The explicit rule is:
[tex]\[ f(n) = 8 \cdot 9^{(n-1)} \][/tex]
3. Identifying the True Statements:
Given the statements:
- The common ratio of the geometric sequence is 8.
- The common ratio of the geometric sequence is 9.
- The explicit rule for the geometric sequence is [tex]\( 8 \cdot 9^{(n-1)} \)[/tex].
- The explicit rule for the geometric sequence is [tex]\( 9 \cdot 8^{(n-1)} \)[/tex].
First Statement: The common ratio of the geometric sequence is 8.
This is incorrect. The common ratio is [tex]\( 9 \)[/tex].
Second Statement: The common ratio of the geometric sequence is 9.
This is correct. The common ratio is indeed [tex]\( 9 \)[/tex].
Third Statement: The explicit rule for the geometric sequence is [tex]\( 8 \cdot 9^{(n-1)} \)[/tex].
This is correct. The explicit rule for the sequence is [tex]\( 8 \cdot 9^{(n-1)} \)[/tex].
Fourth Statement: The explicit rule for the geometric sequence is [tex]\( 9 \cdot 8^{(n-1)} \)[/tex].
This is incorrect. The explicit rule involves the first term [tex]\( 8 \)[/tex] followed by the common ratio [tex]\( 9 \)[/tex], raised to the power of [tex]\( n-1 \)[/tex], so it cannot be [tex]\( 9 \cdot 8^{(n-1)} \)[/tex].
Thus, the correct statements are:
- The common ratio of the geometric sequence is 9.
- The explicit rule for the geometric sequence is [tex]\( 8 \cdot 9^{(n-1)} \)[/tex].
