Use mathematical induction to prove that each statement is true for every positive integer n. n + 2 > n

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Identify the statement to prove using mathematical induction: For every positive integer \(n\), the inequality \(n + 2 > n\) holds.

Base Case: Verify the statement for \(n = 1\). Substitute \(n = 1\) into the inequality to check if \(1 + 2 > 1\) is true.

Inductive Hypothesis: Assume the statement is true for some positive integer \(k\), that is, assume \(k + 2 > k\) holds.

Inductive Step: Using the inductive hypothesis, prove the statement for \(k + 1\). Show that \((k + 1) + 2 > k + 1\) is true.

Conclude that since the base case is true and the inductive step holds, by mathematical induction, the inequality \(n + 2 > n\) is true for every positive integer \(n\).

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주요 개념

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Mathematical Induction

Mathematical induction is a proof technique used to establish that a statement holds for all positive integers. It involves two steps: proving the base case (usually for n=1) and then proving the inductive step, where assuming the statement is true for n=k leads to it being true for n=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 confirms the statement holds at the starting point, providing a foundation for the inductive step.

Inductive Step

The inductive step requires assuming the statement is true for an arbitrary positive integer n=k (inductive hypothesis) and then proving it is true for n=k+1. This step shows the property holds for the next integer, completing the induction process.