1 · IB Maths AA
Proof by induction
Step through the base case and inductive step for three statements, watching the dominoes and the algebra together. Remove a step to see why both are needed.
Try this
- Choose “Remove base case”, go to step 5 and press Play: why does no domino fall even though P(k) ⇒ P(k+1) is proved?
- For the sum of odd numbers, set k = 4 at step 4: count the gold L-shaped layer and check it equals 2k + 1.
- For 5ⁿ − 1, compare the “= 5 × … + 4” line on the canvas with the algebra 5(4m + 1) − 1 for several k.
Exam tip: write “Assume P(k) is true for some k ∈ ℤ⁺”, show clearly the line where you use that assumption, and end with a conclusion that mentions both the base case and the inductive step.