Mathematical induction and recurrence relations from "summary" of Pre-Calculus Know-It-ALL by Stan Gibilisco
Mathematical induction and recurrence relations are methods of proof and analysis that use a sequence of logical steps to arrive at a conclusion. These techniques can provide a more efficient way to solve many types of problems, as well as provide useful insights into the structure of the problem at hand.- The strategy of mathematical induction can be used in proving theorems or solving recurrence relations. A recurrence relation is a sequence of numbers that are determined by formula with each term depending on one or more of the terms preceding it.
- Mathematical induction is an efficient process that allows us to break down a problem and prove it mathematically. It involves showing that an argument works for the initial case, and then assuming that it continues to work for all the subsequent cases.
- Proving that something holds true using mathematical induction can require multiple advances, but if successful, it generally provides a simpler proof than other methods might be able to yield.
- In order to use mathematical induction on recurrence relations, you have to show that your algorithm is true for the base cases, and then assume that it's true for all nth cases. With this assumption, you can solve the equation for the (n+1)th case, which will then have to be true for all n+1.
- With the aid of mathematical induction, complex problems can be broken down into simpler cases until we reach a general rule which states what should hold true for all the cases.
