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Write the explicit rule represented by each geometric sequence. Select all correct statements below.

1.
\begin{tabular}{|c|c|c|c|c|c|}
\hline
[tex]$n$[/tex] & 1 & 2 & 3 & 4 & 5 \\
\hline
[tex]$f(n)$[/tex] & 8 & 72 & 648 & 5832 & 52,488 \\
\hline
\end{tabular}

A. The common ratio of the geometric sequence is 8.
B. The common ratio of the geometric sequence is 9.
C. The explicit rule for the geometric sequence is [tex]$8(9)^{n-1}$[/tex].
D. The explicit rule for the geometric sequence is [tex]$9(8)^{n-1}$[/tex].

Sagot :

GinnyAnswer
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].
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