In Exercises 25–34, use mathematical induction to prove that each statement is true for every positive integer n. n + 2 > n

Key Concepts

Mathematical Induction

Mathematical induction is a proof technique used to establish the truth of an infinite number of statements, typically involving positive integers. It consists of two main steps: the base case, where the statement is verified for the initial value (usually n=1), and the inductive step, where one assumes the statement holds for n=k and then proves it for n=k+1. This method is essential for proving statements that are formulated in terms of integers.

Inequalities

Inequalities are mathematical expressions that show the relationship between two values when they are not equal. In this context, the inequality n + 2 > n indicates that adding a positive number (2) to n will always yield a result greater than n itself. Understanding how to manipulate and interpret inequalities is crucial for proving statements involving them.

Base Case

The base case is the initial step in a mathematical induction proof where the statement is verified for the smallest positive integer, typically n=1. Establishing a true base case is critical because it serves as the foundation for the inductive step, ensuring that the proof can be extended to all positive integers. Without a valid base case, the entire induction process fails.