You can access the assessment and the solutions from the following links:
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Mathematical Induction Proof
Problem Statement: Use mathematical induction to show that a given equation holds for all nonnegative integers ( n ).
Solution: The proof involves two main steps:
- Base Case: Verify the equation for the initial value (usually ( n = 0 ) or ( n = 1 )).
- Inductive Step: Assume the equation is true for an arbitrary integer ( k ), and then show it holds for ( k + 1 ).
The details of the proof can be found in the
Q1.javafile. -
Equation Validation Program
Problem Statement: Write a computer program to validate the given equation.
Solution: A Java program is provided that takes an integer ( n ) as input and verifies whether the equation holds. The code can be found in
Q2.java. -
Lucas Numbers Generator
Problem Statement: Write a recursive method to generate Lucas numbers.
Solution: A recursive Java method is implemented to generate Lucas numbers. The Lucas numbers are similar to Fibonacci numbers but with different initial values. The implementation can be found in
Q3.java. -
Least Airfare Route Finder
Problem Statement: Write a computer program to find a route with the least total airfare that visits each of the cities in a given graph, where the weight on an edge represents the least price available for a flight between the two cities.
Solution: The solution involves implementing an algorithm to find the shortest path in a weighted graph. The code for finding the minimum airfare route can be found in
Q4.java.
Q1.java- Contains the detailed proof for the mathematical induction problem.Q2.java- Java program for validating the equation.Q3.java- Java program for generating Lucas numbers.Q4.java- Java program for finding the least total airfare route.