What is the efficacy class of St? i=1 O O(n) O 0(1) O O(logn) O On*n) O O(n logn)
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![### Complexity Analysis of Summation
#### Question:
What is the efficacy class of \(\sum_{i=1}^{n} \frac{i}{n}\)?
#### Answer Choices:
- \(O(n)\)
- \(O(1)\)
- \(\Theta(\log n)\)
- \(O(n^2)\)
- \(O(n \log n)\)
### Explanation:
Understanding the time complexity for a given algorithmic function requires in-depth analysis of the function itself.
Consider the summation:
\[
\sum_{i=1}^{n} \frac{i}{n}
\]
This can be simplified to:
\[
\frac{1}{n} \sum_{i=1}^{n} i
\]
The arithmetic series sum \(\sum_{i=1}^{n} i\) formula is given by:
\[
\sum_{i=1}^{n} i = \frac{n(n+1)}{2}
\]
Substituting this into our equation, we have:
\[
\frac{1}{n} \cdot \frac{n(n+1)}{2} = \frac{n+1}{2}
\]
For large values of \(n\), the term \(\frac{n+1}{2}\) is asymptotically equivalent to \(O(n)\) since constant factors and lower-order terms are ignored in Big-O notation.
Thus, the efficacy class or time complexity is:
**Answer: \(O(n)\)**](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc88a4be5-ce0c-4e25-871d-a0bd16127b90%2F585fb648-b5b6-43e4-b60a-2332e01c5982%2Fr0y2pm_processed.jpeg&w=3840&q=75)
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