Practice question · True or false
An inductive step alone, without a base case, is enough to prove a statement for all n.
Hints
- Imagine a valid step attached to a false claim.
- Dominoes spaced correctly still stand if nobody pushes the first.
Show the answer
False
Why
False. The claim 'n = n + 1' admits a perfectly valid inductive step, add 1 to both sides, yet is false for every n. Without a base case the implications form a chain anchored to nothing.
Practise Mathematical Induction
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More questions on Mathematical Induction
- Select every statement that is TRUE about mathematical induction.
- Order the parts of a proof by induction that 1 + 2 + ... + n = n(n+1)/2.
- Using the formula 1 + 2 + ... + n = n(n+1)/2, compute the sum of the first 10 natural numbers.
- Complete the statement of the inductive step.
- An induction proof assumes the very statement it is trying to prove holds for n, then proves it for n+1. Why…
- A famous fake proof shows all horses are the same colour by induction: true for one horse, and any n+1 horses…