Exam Questions - Recurrence relationships
Exam Questions - Recurrence relationships
Understanding the Concept
- Recurrence relationships, also known as recursive sequences, are sequences where each term is generated or calculated using values from previous terms.
- The first order recurrence relationship takes the form of u_(n+1) = f(u_n), where the next term depends on the current term. If a first order recurrence relationship is linear, it can usually be solved directly.
- The second order recurrence relationship takes on the form u_(n+2) = f(u_n, u_(n+1)), where the next term depends on the current term and the term before it.
Essential Skills
- The key skill in solving recurrence relationship questions is to understand how to exactly use the recurrence formula.
- Practice how to find the closed form solution, if it exists, from a given recurrence relationship.
- It is crucial to spot and work with patterns in the sequence of values, as these can often simplify the process of solving the problem or even lead directly to the solution.
Key Points to Remember
- Understanding the initial conditions is very important - without the correct initial conditions, you won’t get the right sequence.
- For first order relationships, the equation u_(n+1) = f(u_n) generally requires knowledge of one initial value (u_0 or u_1).
- For second order relationships, the equation u_(n+2) = f(u_n, u_(n+1)) generally requires knowledge of two initial values (u_0, u_1 or u_1, u_2).
- The order of a recurrence relationship refers to the highest number of steps back that a term in the sequence depends on.
Example Questions
- Try out problems that require you to determine a few terms in the sequence given a first or second order recurrence relationship.
- Drill with problems where you are given the recurrence formula and asked to find a specific term in the sequence.
- Challenge yourself with problems that require finding a closed form solution for the sequence where possible.
Related Topics
- Sequences and Series are directly related to recurrence relationships as both involve lists of numbers that follow a certain rule.
- Knowledge of functions can be highly useful in understanding the functions f in the recurrence formulas.
- The topic of induction is also related as it entails proving statements by showing they hold for a base case and the next case relies on the previous one, much like how recurrence relationships work.
Extra Study
- Always revise and solve a diverse range of problems to equip yourself with a strong conceptual foundation.
- To complement your practice, you may look for additional resources such as textbooks, videos, or online tutorials that offer various methods and techniques to solve recurrence relationships.
- Always believe in the power of persistent practice, patience and perseverance while studying recurrence relationships.