Problem 2 - Algebra and Trigonometry

Question

Which of the following is true about Step 2 in a proof by mathematical induction? (i) We prove "P \((k+1)\) is true." (ii) We prove "If \(P(k)\) is true, then \(P(k+1)\) is true."

Step-by-Step Solution

Answer

The correct option is (ii): We prove "If \( P(k) \) is true, then \( P(k+1) \) is true."

Step 1: Understand Mathematical Induction

Mathematical induction is a method of mathematical proof that is commonly used to establish that a given statement is true for all natural numbers. It consists of two main parts: the base case and the induction step.

Step 2: Identify Step 2 in Induction

Step 2 in a proof by mathematical induction is usually referred to as the 'inductive step'. This is where we show that if the statement holds for an arbitrary natural number \(k\), then it must also hold for \(k+1\).

Step 3: Analyze the Given Options

We need to choose between two options: (i) We prove \(P(k+1)\) is true. (ii) We prove "If \(P(k)\) is true, then \(P(k+1)\) is true."Step 2 focuses on showing the logical implication from \(P(k)\) to \(P(k+1)\), rather than directly asserting \(P(k+1)\) is true.

Step 4: Conclusion on Step 2

The inductive step, which is Step 2, involves showing the implication: "If \(P(k)\) is true, then \(P(k+1)\) is true." This is necessary to prove that the statement is true for all subsequent natural numbers after the base case.

Key Concepts

Proof by Induction

Proof by induction is a powerful method used to prove that a statement is true for all natural numbers. It is like a domino effect – if one domino falls and causes the next to fall, all will eventually fall. The process begins by proving that a statement is true for a starting number, often the smallest natural number. Then, we show that if the statement holds for an arbitrary number, it must also hold for the next number. By doing this, we establish that the statement is true for all natural numbers, just like knocking down all dominos by ensuring one falls after the other.

Inductive Step

The inductive step is a crucial part of the proof by induction. It is like connecting the dots from one point to the next in a chain reaction. The goal here is to show that if the statement is true for an arbitrary natural number, say \(k\), it must also be true for \(k+1\). This step involves reasoning from an assumption, known as the "inductive hypothesis". The inductive step typically looks like this: assume the statement holds for \(k\), then logically prove it for \(k+1\). This conditional reasoning confirms that the statement holds consecutively across the natural numbers.

Base Case

In mathematical induction, the base case is the first step in the process. Think of it as setting up the first domino. By showing that the problem holds for the smallest natural number (often 1), we establish a foundation to build on. The base case verifies that the statement is true at the get-go, acting as the starting point for our inductive journey. Without a true base case, the rest of the proof cannot follow. It's essential to demonstrate this step thoroughly to ensure the entire inductive argument stands firm.

Natural Numbers

Natural numbers are the building blocks of proofs by induction. They are the simplest set of numbers: 1, 2, 3, and so on. These numbers are also sometimes called "counting numbers". Understanding proofs intended for all these numbers means comprehending the regularity and order of their progression. In proofs by induction, we start with the smallest natural number in our base case and utilize the structured nature of natural numbers to prove our statements chain from one to the next using the inductive step. Natural numbers make it straightforward to use logical reasoning in induction proofs due to their predictable sequence.