Proof by Induction : How it Works

 

Mathematical induction is a powerful technique in mathematics used to prove statements about natural numbers. It's a method that allows us to establish the truth of an infinite number of propositions, provided we can prove the proposition for a starting point (the base case) and show that if it's true for any particular number, it must also be true for the next number (the induction step).

In this article, we will explore the mechanics of proof by induction, breaking down the base case, the proof for case k, and the proof for case k+1. We'll also delve into a simple example - the sum of the first 'n' natural numbers - and highlight notable algorithms and theorems that have been proved using proof by induction.

Understanding Proof by Induction

Mathematical induction is an invaluable tool when it comes to proving statements that involve natural numbers (positive integers). It operates under the assumption that if a statement is true for some integer 'k,' it must also be true for 'k+1.' This also implies that if the statement is true for ‘k+1’, it holds for ‘k’ as well. The process consists of two main parts:

1. Base Case: In the base case, we prove that the statement is true for the smallest integer value (usually 1). This step establishes the foundation for the induction.
2. Induction Step: In the induction step, we assume the statement holds for a fixed and arbitrary integer 'k' (where 'k' is typically a positive integer). We then use this assumption to prove the statement also holds for 'k+1.' This step extends the statement's truth from 'k' to 'k+1.'

Combining the base case and the induction step allows us to conclude that the statement applies to all positive integers. If the statement holds for ‘k+1’, it will also hold for k+2,k+3 and so on till infinity.

Base Case: Setting the Foundation

The base case is the first crucial step when proving something through proof by induction. It's where we demonstrate that the statement is true for the smallest positive integer, which is typically '1.' That said, let's consider a simple example to illustrate this concept:

Example: The Sum of the First 'n' Natural Numbers

The statement we want to prove is that the sum of the first 'n' natural numbers is denoted using the formula:

1 + 2 + 3 + ... + n = n * (n + 1) / 2

Base Case (n = 1): We begin by proving the statement for the smallest value of 'n,' which is 1.

1 = 1 * (1 + 1) / 2
1 = 1 * 2 / 2
1 = 2 / 2
1 = 1

In this base case, we've shown that the formula holds true when 'n' equals 1.

Proof for Case 'k'

Having established the base case, we move on to the induction step. This is where we assume that the statement is true for a fixed and arbitrary positive integer 'k' and use that assumption to prove the statement for 'k+1.' In our example of the sum of the first 'n' natural numbers, this means we assume that the formula holds for some 'k':

1 + 2 + 3 + ... + k = k * (k + 1) / 2

Now, we use this assumption to show that the formula is also true for 'k+1':

1 + 2 + 3 + ... + k + (k + 1) = (k + 1) * (k + 2) / 2

Step 1: Start with the assumption:

1 + 2 + 3 + ... + k = k * (k + 1) / 2

Step 2: Add 'k + 1' to both sides of the equation:

1 + 2 + 3 + ... + k + (k + 1) = k * (k + 1) / 2 + (k + 1)

Step 3: Factor out 'k + 1' on the right side:

(1 + 2 + 3 + ... + k) + (k + 1) = (k + 1) * (k/2 + 1)

Step 4: Use the assumption for the left side:

(k * (k + 1) / 2) + (k + 1) = (k + 1) * (k/2 + 1)

Step 5: Factor 'k + 1' from both sides:

(k + 1) * (k/2 + 1) = (k + 1) * (k/2 + 1)

Since the left and right sides are equal, we have successfully proven that if the statement is true for 'k,' it's also true for 'k+1.'

We confirmed the base case – the statement holds for k=1 – and that if the statement holds for k, it must hold for k+1 as well. Hence, the statement holds for 1,2,3,4,5 and so on.

Proof for Case 'k+1'

The proof for case 'k+1' is essentially the induction step. We assume that the statement is true for an arbitrary 'k' and use that assumption to prove the statement for 'k+1,' as demonstrated in the previous section.

In the example of the sum of the first 'n' natural numbers, we've shown that if the formula holds for 'k,' it must also hold for 'k+1.' Thus, we can conclude that the formula is true for all positive integers.

Applications of Mathematical Induction

Proof by induction is a versatile technique with applications in various mathematical theorems, algorithms, and problem-solving. Here are eight notable examples where mathematical induction has been employed:

1. Gauss's Formula for the Sum of Natural Numbers: As demonstrated in the example above, the formula for the sum of the first 'n' natural numbers is one of the most well-known applications of mathematical induction.
2. Fibonacci Sequence: Mathematical induction is used to prove properties of the Fibonacci sequence, such as the relationship between consecutive Fibonacci numbers or the sum of the first ‘n’ Fibonacci numbers.
3. Pascal's Triangle: Induction can be used to prove the properties of Pascal's Triangle, a triangular array of binomial coefficients.
4. Divisibility Rules: Proof by induction is used to prove various divisibility rules, such as the rule for determining if a number is divisible by 3 or 9.
5. Inequality Proofs: Mathematical induction is often employed to prove inequalities, such as the AM-GM inequality, which relates the arithmetic mean and geometric mean of a set of positive numbers.
6. Graph Theory: Proof by induction is used in graph theory to prove properties of graphs, including the existence of spanning trees.
7. Recursive Algorithms: Proof by induction is used to analyze recursive algorithms' correctness and time complexity, such as the Tower of Hanoi problem.
8. Combinatorial Identities: Many combinatorial identities, such as binomial coefficient identities and the inclusion-exclusion principle, are proven using proof by induction.

Wrapping Up

Proof by induction is a fundamental technique in mathematics, serving as a reliable method to prove statements about natural numbers. At the end of the day, by establishing the base case and demonstrating the induction step, mathematicians can confidently extend the truth of a statement from a specific case to a universal one.