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

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Step 1: Understand the principle of mathematical induction. It involves two main steps: the base case and the inductive step.

Step 2: Base Case: Verify the statement for the initial value, usually n = 1. Substitute n = 1 into the inequality: 1 + 2 > 1. Check if this is true.

Step 3: Inductive Hypothesis: Assume the statement is true for some positive integer k, i.e., assume k + 2 > k.

Step 4: Inductive Step: Prove the statement for n = k + 1 using the inductive hypothesis. Substitute n = k + 1 into the inequality: (k + 1) + 2 > k + 1.

Step 5: Simplify the inequality from the inductive step: k + 3 > k + 1. Show that this inequality holds true, thus completing the proof by induction.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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