1.7 Proof by Induction 2

🔑 What is Mathematical Proof?

Mathematical proof requires certainty that a relationship is true (or false) for all values being considered, without exception. While we might observe a pattern in a sequence like $1, 2, 4, 8, 16, \ldots$ where terms increase by a factor of 2 and appear to follow $u_n = 2^n$ for $n \geq 0$, observation alone is not proof.

Mathematical induction provides a rigorous method to establish such results with certainty. It involves:

• Setting up a mathematical inductive process using the first term (base case)
• Using a general term to form a convincing argument for all values in an interval

📌 Key Terminology

Mathematical induction: A method of mathematical proof typically used to establish that a given statement is true for all natural numbers.

Conjecture: A mathematical statement which appears to be true, but has not been formally proven.

Base case ($P_1$ or $P_0$): The first step of an inductive proof, showing the statement holds for the initial value (e.g., $n=1$ or $n=0$).

Inductive step: The step in which you show that if the statement holds for $n=k$, it must also hold for $n=k+1$.

$P_k$: The statement being tested for a specific value $k$.

$P_{k+1}$: The statement for the next value in the sequence, $k+1$.