Induction Principle
Prove P(1) and P(k)→P(k+1).
Key idea
Prove P(1) and P(k)→P(k+1).
In depth
To prove P(n) for all n ≥ n₀: prove the base case, and prove that P(k) implies P(k + 1). Like an infinite line of dominoes, each one falls because the previous one did.
Worked example
Prove 1 + 3 + ... + (2n − 1) = n²: base n = 1: 1 = 1. Step: n² + (2n + 1) = (n + 1)².
Real-world uses
- Proving that a loop or a recursive algorithm works for all sizes of input
Common mistake
Assuming P(k + 1) in the inductive step, or leaving out the base case.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Induction Principle?
Prove P(1) and P(k)→P(k+1). It is a topic at the advanced level in Discrete Math & Number Theory.
Where is Induction Principle used in real life?
Real-world examples: Proving that a loop or a recursive algorithm works for all sizes of input.
What should I learn before Induction Principle?
Make sure you are comfortable with Proof by Cases, the previous topic in Discrete Math & Number Theory.
What is a common mistake with Induction Principle?
Assuming P(k + 1) in the inductive step, or leaving out the base case.
How can I practise Induction Principle?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Induction Principle?
Move on to Strong Induction next.
Related advanced topics in Discrete Math & Number Theory
- Tautologies and Contradictions – Always true or always false statements.
- Logical Equivalence – Statements with identical truth tables.
- De Morgan’s Laws – Negation swaps AND and OR.
- Converse and Contrapositive – A conditional equals its contrapositive.
- Quantifiers – “For all” and “there exists”.
- Negating Quantified Statements – Flip the quantifier and negate the predicate.