To describe the sequence given by the terms –272, \(\frac{272}{3}\), \(-\frac{272}{9}\), ..., we can identify the pattern in the sequence and then formulate the general term \(a_n\).
Let's analyze the terms:
- The 1st term is \(-272\).
- The 2nd term is \(\frac{272}{3}\).
- The 3rd term is \(-\frac{272}{9}\).
Notice the following patterns:
- The absolute value of the terms is based on powers of 3:
- The absolute value of the 1st is 272 (which can be considered as \(272 \times \frac{1}{1}\)),
- The absolute value of the 2nd is \(\frac{272}{3}\) (which is \(272 \times \frac{1}{3}\)),
- The absolute value of the 3rd term is \(-\frac{272}{9}\) (which is \(272 \times \frac{1}{9}\)).
The denominators (1, 3, 9) are powers of 3 (\(3^0, 3^1, 3^2\)).
Also, it alternates signs between positive and negative. Noting this, the general term can be described as follows:
- The sign changes based on \(n\):
- For odd \(n\), the term is negative: \((-1)^{n}\)
- For even \(n\), the term is positive.
Putting this together, we can formulate the general term \(a_n\) of the sequence as:
\[ a_n = (-1)^{n} \times \frac{272}{3^{n-1}} \]
This equation effectively reflects the alternating signs and decreasing magnitudes of the terms in the sequence.
So, the final answer in the required format is:
\[ a_n = (-1)^{n} \cdot \frac{272}{3^{n-1}} \]
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