To determine whether the sequence \( 81, 72, 63, 54, 45, \ldots \) is arithmetic or geometric, we will first check if it is arithmetic. A sequence is arithmetic if there is a constant difference between consecutive terms.
Let's calculate the differences between the consecutive terms:
- \( 72 - 81 = -9 \)
- \( 63 - 72 = -9 \)
- \( 54 - 63 = -9 \)
- \( 45 - 54 = -9 \)
The common difference is \( -9 \), and since the difference is constant, this sequence is an arithmetic sequence with a common difference of \( -9 \).
Next, we can conclude that this sequence is not geometric because a geometric sequence would require a constant ratio between consecutive terms.
To verify if it's geometric, let's calculate the ratios:
- \( \frac{72}{81} = \frac{8}{9} \)
- \( \frac{63}{72} = \frac{7}{8} \)
- \( \frac{54}{63} = \frac{6}{7} \)
- \( \frac{45}{54} = \frac{5}{6} \)
Since the ratios are not constant, the sequence is not geometric.
In summary, the sequence is arithmetic with a common difference of \( -9 \).
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