The Principle of Mathematical Induction
The Principle of Mathematical Induction is a method used to prove that a statement is true for all natural numbers. It involves two steps: 1. **Base Case**: Show that the statement holds for the first natural number, usually 1. 2. **Inductive Step**: Assume it’s true for a number \( n \) (this is called the inductive hypothesis), and then prove it also holds for \( n + 1 \). If both steps are successful, it implies the statement is true for all natural numbers, building on the idea that if it holds for one number, it holds for the next one too.
Additional Insights
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Mathematical induction is a proof technique used to show that a statement is true for all natural numbers. It involves two steps: first, proving the statement is true for the initial number (usually 1). Then, assuming it holds for an arbitrary number \( n \), we show it must also be true for \( n+1 \). This creates a chain reaction; if it’s true for 1, then it’s true for 2, then 3, and so on. Induction is like climbing a staircase: once you can step on the first stair and reach the next, you can reach them all!