Problem 28 - Contemporary Precalculus

Question

If the given statement is true, prove it. If it is false, give a counterexample. Let \(q\) be a fixed integer. Suppose a statement involving the integer \(n\) has these two properties: (i) The statement is true for \(n=q\) (ii) If the statement is true for \(n=k\) (where \(k\) is any integer with \(k \geq q\) ), then the statement is also true for \(n=k+1\) Then we claim that the statement is true for every integer \(n\) greater than or equal to \(q\). (a) Give an informal explanation that shows why this claim should be valid. Note that when \(q=1,\) this claim is precisely the Principle of Mathematical Induction. (b) The claim made before part (a) will be called the Extended Principle of Mathematical Induction. State the two steps necessary to use this principle to prove that a given statement is true for all \(n \geq q\). (See the discussion on pages \(866-867\).)

Step-by-Step Solution

Answer

Question: Using the Extended Principle of Mathematical Induction, explain why the following claim is valid: If a statement is true for some fixed integer q and it is also true for any integer k ≥ q such that if it's true for k, then it's true for (k+1), then the statement is true for all integers n ≥ q. Answer: The claim is valid because the Extended Principle of Mathematical Induction allows us to prove a given statement for all integers n ≥ q in two steps: showing the base case and completing the inductive step. By following these steps, we establish that the statement is true for n=q, which serves as the starting point. Then, under the assumption that the statement is true for any integer k ≥ q, we prove that it is also true for the next integer n=k+1. This process continues, showing the statement is true for all consecutive integers n ≥ q, which is why the claim is valid.

Step 1: (a) Informal explanation of the claim's validity

The claim states that if a statement is true for some fixed integer \(q\) and it is also true for any integer \(k \geq q\) implying the statement is true for \((k+1)\), then the statement is true for all integers \(n \geq q\). This claim is essentially an extension of the Principle of Mathematical Induction. In the Principle of Mathematical Induction, we base our claim on the following steps: 1. If the statement is true for a base case (usually \(n=1\)). 2. If we assume the statement is true for \(n=k\) implies that it is also true for \(n=(k+1)\). Using these steps, we can establish that the statement holds for all values of \(n \geq 1\). In this case, our claim replaces the base case of \(n=1\) with a fixed integer \(q\), and we are trying to prove the statement for all \(n \geq q\). The intuition behind this claim's validity is as follows: Since the statement is true for \(n=q\), and if we assume that it is true for \(n=k\) such that \(k \geq q\), then based on the given condition, the statement must also be true for \(n=k+1\). This process continues, and every time we show the statement is true for an integer \(n \geq q\), we can also show it is true for the next integer. As a result, we can conclude that the statement is true for all integers \(n \geq q\).

Step 2: (b) Extended Principle of Mathematical Induction and the two necessary steps for the proof

Using the Extended Principle of Mathematical Induction, we can prove a given statement for all integers \(n \geq q\) with the help of the following two steps: 1. Base Case: Show that the statement is true for the fixed integer \(n=q\). 2. Inductive Step: Assume that the statement is true for an arbitrary integer \(k \geq q\). Then, prove that the statement is also true for the next integer \(n=k+1\). By completing both of these steps, we will have provided sufficient evidence to support the claim that the statement must be true for all integers \(n \geq q\) using the Extended Principle of Mathematical Induction.

Key Concepts

Mathematical Induction

Mathematical induction is a powerful proof technique used to establish that a statement is true for all natural numbers. It is a bit like a row of dominoes. You knock over the first domino (proving the base case), and the dominos keep falling (proving the inductive step ensures all subsequent cases hold). This method lets you prove a seemingly endless number of statements with just a few steps.

• It generally applies to statements involving integers.
• The steps involve establishing an initial truth and then using an assumption step to prove further truths.

The beauty of mathematical induction lies in its simplicity and thoroughness. By merely assuming a little (the base case and the inductive step), we can prove a lot!

Base Case

The base case in mathematical induction is where everything begins. It's the starting point of our proof journey. By demonstrating that a statement holds true for an initial value, often for the smallest value under consideration, like 1, we lay down the first stone.

• It's akin to proving the first domino in a sequence will fall when pushed.
• Without a correctly established base case, the entire proof could topple.

Checking the base case ensures that our logical starting point is sound. Once validated, we can focus on the inductive step with confidence.

Inductive Step

The inductive step is where the magic of mathematical induction truly happens. After establishing the base case, we assume the statement is true for an arbitrary natural number, known as "assuming for the case of \(n = k\)". Then, we must prove that the statement also holds for the next number, \(n = k + 1\).

• This step bridges the gap between individual instances of the statement.
• It ensures that if the statement holds for one number, it cascades to the next.

The inductive step is crucial—it is what turns finite processes into infinite proofs, covering all numbers in sequence.

Proof by Induction

Proof by induction is like a logical chain reaction, starting with a few well-placed links (the base case and inductive step) that trigger a sequential validation for all integers beyond the start point. It grants mathematicians a tool to prove intricate properties of numbers with ease and elegance.

• Combines a base starting point with a domino effect of proof.
• Relies on logical continuity between consecutive numbers.

When executed correctly, proof by induction illustrates the consistency and correctness of mathematical statements, reinforcing the confidence in their universal truth.