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Chapter 8: Problem 31

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

Short Answer

Expert verified
By employing mathematical induction, it has been proven that for every positive integer n, n+2 > n. The base case was found to be true and the inductive step confirmed the truth of the statement for all positive integers.

Step by step solution

01

Base Case

In the case of the base, n=1. Plug 1 into the given expression and validate the given condition: 1 + 2 > 1. This results in 3 > 1, which is a true statement. Therefore, the base case holds for the initial integer n=1.
02

Inductive Hypothesis

Make an assumption that the statement is true for some positive integer k, i.e., k+2 > k.
03

Inductive Step

Prove the statement is true for the next integer, i.e., for (k+1). Consider the expression (k+1) + 2. This can be simplified as k + 3 which is definitely greater than (k+1). Therefore, assuming the statement holds for some positive integer k, it also holds for the next positive integer (k+1).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Base Case
In mathematical induction, the base case is the starting point. It establishes the truth of a statement for the initial value. Let's explore this concept using our example statement:
  • We want to prove that for every positive integer \(n\), \(n+2 > n\).
  • The base case checks this assertion for \(n = 1\). By substituting \(n = 1\) into the inequality, we evaluate: \(1 + 2 > 1\).
  • This simplifies to \(3 > 1\), a true statement.
By proving the base case, we set the stage for our induction process. This foundation ensures that the statement is true, at least starting from \(n = 1\). All further steps build off this truth, confirming it holds for all integers greater than this base.
Inductive Hypothesis
The inductive hypothesis is a crucial element of mathematical induction. It involves making an assumption that a statement holds true for a particular integer, \(k\). Let's break it down:
  • The hypothesis assumes: if \(k+2 > k\) is true, then we will use this assumption to aid in proving the next step.
  • This step isn't a proof on its own, but rather an assertion that sets up the next part of the induction.
The inductive hypothesis allows us to move from the base case to a more generalized situation. It acts like a temporary truth we use to logically carry our argument forward. Even though it's an assumption, it is critical to the mathematical induction process, allowing us to connect different cases together.
Inductive Step
The inductive step is where we use our assumption from the inductive hypothesis to prove that if the statement holds for \(k\), it must also hold for \(k+1\). This step involves:
  • First, consider \((k+1)+2\) or \(k+3\).
  • We need to show that \(k+3 > k+1\).
  • This easily simplifies to \(2 > 0\), confirming the truth of the statement.
During this step, we transition from the specific case established in the base and inductive hypothesis to the broader case of \(k+1\). Successfully proving this step confirms the statement holds for all successive integers starting from our base case. The completion of this step ensures the integrity of the induction, proving the statement's truth for every positive integer \(n\).

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