This segment demonstrates the back substitution method to solve the recurrence relation. The speaker systematically substitutes the recurrence relation into itself, revealing a pattern that leads to a general formula for the time complexity. This illustrates a powerful technique for solving recurrence relations, particularly useful for understanding the underlying mathematical principles. The initial recurrence relation is deemed too complex for easy solution. The speaker then explains the process of simplifying it using Big O notation, focusing on the dominant terms to arrive at a more manageable and solvable form, highlighting the practical application of asymptotic analysis in algorithm analysis. The speaker meticulously breaks down a recursive function, analyzing the time complexity of each component (loop, conditional statement, recursive call). This detailed step-by-step approach culminates in the derivation of the recurrence relation, showcasing a clear and methodical problem-solving process. This video demonstrates two methods for solving recurrence relations to determine algorithm time complexity. The first method uses a recursive tree to visualize the algorithm's execution and sum the time taken at each step. The second method uses back substitution (or induction) to iteratively substitute the recurrence relation until a closed-form solution is obtained. Both methods are applied to a sample recursive function, yielding a time complexity of O(n²).