Problem 35 - College Algebra

Question

Explain how to use mathematical induction to prove that a statement is true for every positive integer \(n\).

Step-by-Step Solution

Answer

The induction process involves starting with proving a chosen statement for a base case (often \(n=1\)), then considering an arbitrary case \(k\), and assuming the statement is true. Following that, demonstrate that if it's true for \(k\), then it should also be true for \(k+1\). These steps prove the validity of the statement for all positive integers.

Step 1: Basis Step

The basis or base step needs to establish the result for the first case, most commonly when \(n=1\). The goal is to prove that the statement holds true for \(n=1\).

Step 2: Inductive Hypothesis

Assume that the statement is true for some positive integer \(k\). This assumption is called the Inductive Hypothesis. It's not proven at this point, it's assumed to hold, and it's from there that we proceed to the next step.

Step 3: Inductive Step

Using the inductive hypothesis, demonstrate that the statement is true for \(k+1\). Essentially, it's required to show that if the statement is presumed to be true at some step, it has to be true for the next step as well.

Key Concepts

Basis Step

The basis step of mathematical induction is the foundational part of the proof. Think of it as setting the stage. Without a firm foundation, the entire structure might not hold. In this step, we aim to establish that the given statement holds for the beginning of our sequence, usually when \(n=1\). This example value, such as \(n=1\), is the simplest case to verify.

Here's how you can think about it:

• Choose the smallest value of \(n\), often \(n=1\).
• Substitute this value into the statement or formula you're trying to prove.
• Show that the statement is true, which could involve simple arithmetic or logic.

By confirming this initial case, you lay down the first brick in your proof. Every induction proof begins with this step to ensure the statement is grounded in reality before moving forward.

Inductive Hypothesis

The inductive hypothesis is a critical assumption made during the process of mathematical induction. It’s the bridge between the known and the unknown. Once we've shown the statement is true for the base case, we then hypothesize, or assume, that it is true for some arbitrary positive integer \(k\). This is like saying, "Let's temporarily assume I've climbed to the \(k\)th step of a staircase."

It's important to understand:

• You do not provide a proof here; it's purely an assumption.
• This hypothesis is what allows us to make the next critical step in the proof.

While the inductive hypothesis is assumed true, it's this assumption that helps us progress to proving the statement for the next step, \(k+1\). It’s a temporary 'believing-without-seeing' to eventually ensure the truth for the subsequent number.

Inductive Step

The inductive step is where the magic of mathematical induction happens. This part is about proving that if we assume our statement is true for \(n=k\), then it must also be true for \(n=k+1\). Think of this step as building a new staircase step; if you've safely climbed to the \(k\)th step, this shows you how to reach \(k+1\).

Main things to consider:

• Start with the inductive hypothesis: assume the statement is valid for \(k\).
• Use this assumption to show the statement's validity for \(k+1\). This typically involves algebraically modifying the original statement.
• This step confirms the logical progression from \(n=k\) to \(n=k+1\).

Successfully completing the inductive step means your proof covers all positive integers, due to two fundamentals: you've shown it works for the first one (basis step), and the transition from any number \(k\) to the next \(k+1\). This is why induction is so powerful; it's like dominos—if one falls, all will follow!