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2. Solve the equation T(n) = T(n-1) + n. The base condition is T(0) = 0.

2. Solve the equation T(n) = T(n-1) + n. The base condition is T(0) = 0.

Operations Research : Applications and Algorithms
Operations Research : Applications and Algorithms
4th Edition
ISBN:9780534380588
Author:Wayne L. Winston
Publisher:Wayne L. Winston
Chapter20: Queuing Theory
Section20.4: The M/m/1/gd/∞/∞ Queuing System And The Queuing Formula L = Λw
Problem 14P
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Please help me with Question 2. Please provide a step-by-step explanation of the solution as I am really struggling with this.
Introduction to Recurrences A recurrence is defined as being an equation or inequality that describes a function in terms of its value on smaller inputs. For this exercise, we will write an equation for the function T(n) to achieve the given sequence. State any initial conditions for the numbers provided. The initial condition of T(0) = 3. T(n) = 2T(n − 1) To solve this equation, we need to solve for T(n − 1). That can be solved by evaluating T(n − 1) = 2T ((n − 1) − 1). We can plug this into our original equation for T(n): = 2(2T(n-1-1)) We will do the same thing again for the third iteration: = 2(2(2T(n-1-1. -1))) Since we can see an emerging pattern, we can try this where the value k represents the current iteration (e.g. k = 3 for the third call of T(n) in this case): = 2k T(n - k) We will continue to do this until n - k in the function T(n - k) reaches 0. This will happen when k = n given that n - n = 0. Thus, our final equation can be written as: = 2" T(0) Since we have been given a base case to solve, which is T(0) = 3, we can now solve this for any value of n: = 2n (3) Practice Problems 1. Solve the equation T(n) = 2n - T(n − 1). The base condition is T(0) = 1. 2. Solve the equation T(n) = T(n − 1) + n. The base condition is T(0) = 0.
Transcribed Image Text:Introduction to Recurrences A recurrence is defined as being an equation or inequality that describes a function in terms of its value on smaller inputs. For this exercise, we will write an equation for the function T(n) to achieve the given sequence. State any initial conditions for the numbers provided. The initial condition of T(0) = 3. T(n) = 2T(n − 1) To solve this equation, we need to solve for T(n − 1). That can be solved by evaluating T(n − 1) = 2T ((n − 1) − 1). We can plug this into our original equation for T(n): = 2(2T(n-1-1)) We will do the same thing again for the third iteration: = 2(2(2T(n-1-1. -1))) Since we can see an emerging pattern, we can try this where the value k represents the current iteration (e.g. k = 3 for the third call of T(n) in this case): = 2k T(n - k) We will continue to do this until n - k in the function T(n - k) reaches 0. This will happen when k = n given that n - n = 0. Thus, our final equation can be written as: = 2" T(0) Since we have been given a base case to solve, which is T(0) = 3, we can now solve this for any value of n: = 2n (3) Practice Problems 1. Solve the equation T(n) = 2n - T(n − 1). The base condition is T(0) = 1. 2. Solve the equation T(n) = T(n − 1) + n. The base condition is T(0) = 0.
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