List any two applications of recurrence relations.

Recurrence relations are widely used in computer science and mathematics to model and solve problems involving sequences and patterns. Here are two key applications:

1. Algorithm Analysis Recurrence relations are essential in analyzing the time complexity of recursive algorithms. For example, the recurrence relation for the merge sort algorithm is: $$T(n)=2T\left(\frac{n}{2}\right)+O(n)$$ This relation helps determine how the runtime scales with input size, enabling optimization and comparison of algorithms.

2. Computer Graphics and Animation Recurrence relations are used in rendering fractals, simulating natural phenomena (e.g., tree growth, flame algorithms), and calculating trajectories in physics-based animations. For instance, the Mandelbrot set is defined using a recurrence relation: $$z_{n+1}=z_n^2+c$$ where $z_0=0$ and $c$ is a complex constant. This relation generates intricate fractal patterns used in visual effects.