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Mathematical Induction is a powerful proof technique used in mathematics to show that a given statement is true for all natural numbers. It is often applied to formulas, inequalities, divisibility properties, and problems involving sequences or series.

The principle of induction works in two main steps:

Base Case: Verify that the statement is true for the first natural number, usually
𝑛=1
n=1. This establishes the starting point of the proof.
Example: Prove that
1=1
1=1.

Induction Step: Assume that the statement is true for some arbitrary natural number
𝑛=𝑘
n=k (called the induction hypothesis). Then, prove that the statement also holds true for
𝑛=𝑘 + 1
n=k+1. If this step is valid, it shows that the truth of the statement for one case guarantees the truth of the next.

By completing both steps, we conclude that the statement is true for all natural numbers.

Mathematical induction is like a domino effect: once the first domino falls (base case) and each domino is shown to knock over the next one (induction step), then all dominos will fall.

This method is widely used in computer science, algorithms, and discrete mathematics for proving correctness and properties of functions.
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