To express the series \(100 + 90 + 81\) using sigma notation, we can identify a pattern in the terms.
- Identifying the pattern:
- The first term is \(100\).
- The second term is \(90\) (which is \(100 - 10\)).
- The third term is \(81\) (which is \(90 - 9\) or \(100 - 19\)).
The pattern can be seen as a decreasing sequence: the first term is \(10^2\), the second term is \(9^2\), and the third term is \(8^2\). We can generalize this to include the squares of decreasing integers starting from \(10\).
- Writing in sigma notation: The series can be expressed in the following way:
\[ \sum_{n=8}^{10} n^2 \]
This notation indicates that we are summing the squares of the integers from \(8\) to \(10\), which corresponds to the terms \(8^2 + 9^2 + 10^2 = 64 + 81 + 100\), but we need a correction to map correctly to the series we are actually summing.
Instead, noting the terms clearly, the sigma notation for the specific series you provided should reflect the terms themselves:
\[ \sum_{k=0}^{2} (10-k)^2 \]
Here, when \(k=0\), it yields \(10^2 = 100\), when \(k=1\), it yields \(9^2 = 90\), and when \(k=2\), it yields \(8^2 = 81\).
Thus, an accurate sigma notation would be:
\[ \sum_{k=0}^{2} (10-k)^2 \]
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