What are the two conditions required to prove a conjecture using mathematical induction?

Conditions

• The statement is a proposition $P(n)$ indexed by natural numbers.
• The base case is typically checked at $n=1$.
• The inductive step assumes $P(k)$ is true to prove $P(k+1)$.

Reasoning, step by step

1. Verify Condition 1: Prove that $P(1)$ is true.
2. Verify Condition 2: Assume $P(k)$ is true (inductive hypothesis) and prove that $P(k+1)$ follows.
3. Conclude that by the principle of mathematical induction, $P(n)$ is true for all natural numbers n.

Example

The slide lists 'Condition 1 : $P(1)$ is True for $n = 1$' and 'Condition 2 : If $P(n)$ true for $n = k$, then $P(n)$ true for $n = k+1$'. The narrator compares this to a domino effect where the first domino falls and each subsequent one knocks over the next.

Common misconceptions

• Thinking that proving the base case alone is sufficient.
• Confusing the inductive hypothesis (assuming $P(k)$) with the conclusion (proving $P(n)$ for all n).

Watch the explanation

Explore next

• What is the inductive hypothesis?
• How is the base case verified for the sum of even integers?