Ch. 8 - Sequences, Induction, and Probability

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

Blitzer 8th Edition

Ch. 8 - Sequences, Induction, and Probability

Problem 31

Chapter 9, Problem 31

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

Verified step by step guidance

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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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 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.

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