### 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: \(O(n)\)**

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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)

Database System Concepts

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ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

Section: Chapter Questions

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