Mathematical Induction §

Introduction §

Mathematical Induction is a powerful and elegant technique in mathematics, primarily used for proving the validity of a given statement for all natural numbers. It’s akin to a domino effect; if one statement holds true, it triggers a chain reaction ensuring all subsequent statements are true. This method is particularly useful in discrete mathematics, computer science, and when dealing with sequences and series.

Axiom of Mathematical Induction §

A basic axiom:

Suppose that $P(n)$ is a proposition, or property, depending on a positive integer $n\in \mathbb{N}$. Suppose that…

1. $P(1)$ is true, and
2. if $P(k)$ is true, then $P(k+1)$ is true.

Then $P(n)$ is true for all $n\in \mathbb{N}$.

The Process §

Formerly, the axiom is stated in proper form, however, as a process, mathematical induction involves two crucial steps:

1. Base Case: This is the initial step where we prove that the statement is true for the initial value (usually, but not necessarily, for $n=1$).

2. Inductive Step: Here, we assume the statement holds for a certain natural number $k$. Then, we use this assumption to show that the statement must also be true for $k+1$.

By completing these two steps, we effectively prove the statement for all natural numbers.

Example in Practice §

Consider the statement $S(n)$: “The sum of the first $n$ natural numbers is $\frac{n(n+1)}{2}$”. To prove this using induction, we follow these steps:

1. Base Case: For $n=1$, the sum of the first natural number is $1$. The formula gives $\frac{1(1+1)}{2}=1$. Thus, the base case holds.

2. Inductive Step: Assume $S(k)$ is true; i.e., the sum of the first $k$ numbers is $\frac{k(k+1)}{2}$. Now, we need to prove $S(k+1)$ is true. The sum of the first $k+1$ numbers is the sum of the first $k$ numbers plus $k+1$, which is $\frac{k(k+1)}{2}+(k+1)$. Simplifying this, we get $\frac{(k+1)(k+2)}{2}$, thus proving $S(k+1)$ holds.

Other examples may be found in divisibility proofs, etc. Note also that demonstrating the inductive step can be done by showing that the $LHS$ is equal to $RHS$ by any means possible - not necessarily transforming one to the other, often transforming both to the same form.

Conclusion §

Mathematical Induction is a fundamental technique in mathematics, particularly useful for proving propositions or formulas related to natural numbers. It’s a testament to the beauty and logic of mathematics, showing how a simple assumption, if correctly proven, can unravel a universal truth.

---

Test Questions §

1. STARTI [Basic] Question: What are the two main steps involved in a mathematical induction proof? Back: The two main steps are the Base Case, where the statement is proven for the initial value, and the Inductive Step, where the assumption that the statement holds for a certain number is used to prove it for the next number. ENDI
2. STARTI [Basic] Question: In the context of mathematical induction, what does the Base Case aim to prove? Back: The Base Case aims to prove that the statement or proposition is true for the initial value, typically for n = 1. ENDI
3. STARTI [Basic] Question: How is the ‘domino effect’ analogy relevant to mathematical induction? Back: The ‘domino effect’ in mathematical induction refers to the idea that proving the statement for one number (like knocking over the first domino) ensures the truth of the statement for all subsequent numbers (like a chain of falling dominoes). ENDI