Agenda

<aside> 📌 Reminders

• Test 3 November 13, 2024 </aside>

What is Strong Induction?

Strong induction is a proof technique, similar to regular induction, used to establish the truth of a statement $P(n)$ for all integers $n \geq n_0$. However, unlike regular induction, strong induction assumes the truth of $P(k)$ for all $k$ within a certain range to prove $P(n)$.

Steps in a Strong Induction Proof

1. Base Case: Prove that $P(n_0)$ is true for the smallest value of $n$ (often $n_0 = 1$).
2. Inductive Hypothesis: Assume $P(k)$ is true for all values $n_0 \leq k \leq n$.
3. Inductive Step: Use the assumption that $P(k)$ is true for all $k \leq n$ to prove $P(n + 1)$.
4. Conclusion: By the principle of strong induction, $P(n)$ is true for all $n \geq n_0$.

Key Difference: Strong vs. Regular Induction

• Regular Induction: Only assumes $P(n)$ to prove $P(n + 1)$.
• Strong Induction: Assumes $P(k)$ for all $k \leq n$ to prove $P(n + 1)$.

Strong induction is particularly useful for problems involving divisibility, sequences, or recursive definitions.

Practice