Method of Mathematical Induction
🎯 What is Mathematical Induction?
Mathematical Induction is a powerful proof technique used to prove that a statement is true for all natural numbers (or all integers starting from some value). Think of it like a chain of dominoes — if you can knock down the first one, and each domino knocks down the next, then all dominoes will fall!
📐 The Two Steps of Mathematical Induction
Step 1: Base Case (Foundation)
Prove that the statement is true for the first value, usually $n = 1$.
This is like pushing the first domino.
Show that $P(1)$ is true.
Step 2: Inductive Step (Chain Reaction)
Assume the statement is true for $n = k$ (this assumption is called the Inductive Hypothesis).
Then prove it must be true for $n = k + 1$.
Assume $P(k)$ is true → Prove $P(k+1)$ is true.
🔑 Conclusion: If both steps are completed, then the statement $P(n)$ is true for all natural numbers $n \geq 1$.