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Proof by Induction

Proof by induction is a mathematical technique used to prove that a statement is true for all natural numbers. It involves two main steps: first, you show that the statement is true for the initial number (usually 1). Then, you assume it’s true for some number \( n \) and prove that it must also be true for \( n+1 \). If both steps are successful, you can conclude that the statement is true for all natural numbers. This method is often used in mathematics to establish results involving sequences, formulas, or other properties that build on each other.

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    Proof by induction is a mathematical technique used to demonstrate that a statement is true for all natural numbers. It consists of two main steps: 1. **Base Case**: First, you prove the statement for the initial value, usually for 1. This shows it holds true at the starting point. 2. **Inductive Step**: Next, you assume the statement is true for an arbitrary natural number \( n \) and then prove it for \( n+1 \). If both steps are satisfied, you conclude that the statement is true for all natural numbers, as the truth is established starting from the base case and builds onward.