True or false? On the educational website, the text would appear as follows:---\[ \left\lfloor \frac{n}{5} \right\rfloor + 6n + 3 \text{ is } \Theta(n) \]In this equation, \(\left\lfloor \frac{n}{5} \right\rfloor\) represents the floor function applied to \( \frac{n}{5} \), which means rounding down \( \frac{n}{5} \) to the nearest integer. The term \(6n + 3\) is a linear function of \(n\). When combined, the expression \(\left\lfloor \frac{n}{5} \right\rfloor + 6n + 3\) has a growth rate that is linearly proportional to \(n\), and thus it falls within the asymptotic notation \(\Theta(n)\).The \(\Theta(n)\) notation indicates that the function grows linearly with respect to \(n\), meaning that for large values of \(n\), the expression is bound both above and below by linear functions of \(n\), up to constant factors.--- ### Analyzing Arithmetic Series in Terms of Big Theta NotationThe following expression represents an arithmetic series:\[ 3 + 6 + 9 + \ldots + 3n \]The sum of this series can be expressed in Big Theta notation as:\[ \Theta(n^3) \]### ExplanationIn this series, the common difference between consecutive terms is 3. To understand the total sum's complexity, we explore the sum of the first \( n \) terms of this arithmetic series.Given an arithmetic series:\[ a + (a+d) + (a+2d) + \ldots + [a + (n-1)d] \]where:- \( a = 3 \) (the first term)- \( d = 3 \) (common difference)- \( n \) is the number of termsThe sum \( S_n \) of the first \( n \) terms of an arithmetic series is given by the formula:\[ S_n = \frac{n}{2} [2a + (n-1)d] \]Substitute \( a \) and \( d \) into the formula:\[ S_n = \frac{n}{2} [2(3) + (n-1)(3)] \]\[ S_n = \frac{n}{2} [6 + 3n - 3] \]\[ S_n = \frac{n}{2} [3n + 3] \]\[ S_n = \frac{n}{2} \cdot 3(n + 1) \]\[ S_n = \frac{3n^2 + 3n}{2} \]In Big Theta notation, higher-order terms and constants are significant for representing the growth rate. For large values of \( n \), \( S_n \) is dominated by the \( n^3 \) term:\[ S_n = \Theta(n^3) \]Thus, the series \( 3 + 6 + 9 + \ldots + 3n \) can be expressed as \( \Theta(n^3) \) in Big Theta notation, indicating that the complexity or growth rate of the sum is cubic.

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Description: True or false? On the educational website, the text would appear as follows:---\[ \left\lfloor \frac{n}{5} \right\rfloor + 6n + 3 \text{ is } \Theta(n) \]In this equation, \(\left\lfloor \frac{n}{5} \right\rfloor\) represents the floor function appli

Answered: +6n +3 is Θ (n)

Math

Advanced Math

+6n +3 is Θ (n)

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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