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

Verified step by step guidance

1. Identify the statement to prove using mathematical induction: for every positive integer $n$, $n+\; 2\; >n$.
2. Base Case: Verify the statement for $n=\; 1$. Substitute $1$ into the inequality to check if $1\; +\; 2\; >\; 1$ holds true.
3. Inductive Hypothesis: Assume the statement is true for some arbitrary positive integer $k$, that is, assume $k\; +\; 2\; >\; k$ is true.
4. Inductive Step: Using the inductive hypothesis, prove the statement for $k\; +\; 1$. Show that $(k\; +\; 1)\; +\; 2\; >\; (k\; +\; 1)$ holds by simplifying and comparing both sides.
5. Conclude that since the base case is true and the inductive step holds, by mathematical induction, the statement $n\; +\; 2\; >\; n$ is true for every positive integer $n$.

Key Concepts

Mathematical Induction

Mathematical induction is a proof technique used to show that a statement holds for all positive integers. It involves two steps: proving the base case (usually n=1) is true, and then proving that if the statement holds for an arbitrary integer k, it also holds for k+1.

Base Case Verification

The base case is the initial step in induction where the statement is verified for the smallest positive integer, often n=1. This step establishes the starting point for the induction process and ensures the statement is true at the beginning.

Inductive Step

The inductive step assumes the statement is true for some integer k (inductive hypothesis) and then proves it must be true for k+1. This step links each case to the next, creating a chain of truth for all positive integers.