To express the given sequence \( 5, -\frac{5}{2}, \frac{5}{4}, \ldots \) as a geometric series in sigma notation, we first identify the first term and the common ratio.
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First Term (\(a\)): The first term of the sequence is \(5\).
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Common Ratio (\(r\)): We compute the common ratio by dividing the second term by the first term: \[ r = \frac{-\frac{5}{2}}{5} = -\frac{1}{2} \] We can verify this with the next term as well: \[ r = \frac{\frac{5}{4}}{-\frac{5}{2}} = \frac{5}{4} \cdot \frac{-2}{5} = -\frac{1}{2} \] Thus, the common ratio is confirmed to be \( -\frac{1}{2} \).
The general form for a geometric series can be expressed as: \[ a + ar + ar^2 + ar^3 + \ldots + ar^{n-1} \] where \(a\) is the first term, \(r\) is the common ratio, and \(n\) is the number of terms.
For our sequence, we have:
- \(a = 5\)
- \(r = -\frac{1}{2}\)
- We want to sum the first 50 terms, so \(n = 50\).
The sum of the first \(n\) terms can be written in sigma notation as: \[ \sum_{k=0}^{49} 5 \left(-\frac{1}{2}\right)^k \]
Therefore, the geometric series for the given sequence that sums the first 50 terms in sigma notation is: \[ \sum_{k=0}^{49} 5 \left(-\frac{1}{2}\right)^k \]