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Theory of Mathematical Induction
Mathematical induction is a method used to prove that a mathematical statement is true for all positive integers. It involves two main steps:

Step 1: Base Case
Prove the statement is true for n = 1 (or the smallest possible value).

Step 2: Inductive Step
Assume the statement is true for n = k, and then prove it is true for n = k + 1.

If both steps are successfully proven, then the statement is true for all positive integers.

Applications
Mathematical induction is commonly used to prove:

1. Algebraic identities
2. Inequalities
3. Divisibility properties
4. Geometric formulas

Importance
Mathematical induction provides a rigorous method for proving statements about infinite sets of numbers, ensuring the validity of mathematical formulas and theorems.