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). 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.