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Mathematical Induction
Induction proves a base case and then shows that truth at k forces truth at k+1.
How it works
Induction proves a base case and then shows that truth at k forces truth at k+1. Name the quantities before calculating, keep each transformation logically equivalent, and check the result against the original conditions.
Worked example
Outline a proof that 1+…+n=n(n+1)/2.
- Verify n=1.
- Assume the formula for n=k.
- Add k+1 and simplify to (k+1)(k+2)/2.
The base and inductive step establish the formula for all positive integers.
Try it now
These questions come from the same validated skill generator used by the game.
Common mistakes
- Using a remembered formula without identifying what each symbol represents.
- Changing a value or condition during an intermediate step.
- Skipping the final substitution, estimate, or logical check.
Challenge
Prove by induction that 3 divides 4ⁿ−1 for every positive integer n.
Questions learners ask
What is the key idea in mathematical induction?
Induction proves a base case and then shows that truth at k forces truth at k+1.
How should I check an answer?
Substitute it into the original conditions, check units and signs, and decide whether its size is reasonable.