Practice question · Fill in the blanks
Complete the statement of the inductive step.
The inductive step must prove that if the claim holds for n = k, then it also holds for ______.
Word bank: n = 1 · n = k · n = k + 1 · every n
Hints
- The step moves the claim along by exactly one rung.
- Proving 'every n' directly would make induction unnecessary.
Show the answer
The inductive step must prove that if the claim holds for n = k, then it also holds for n = k + 1.
Why
The step proves the claim for n = k+1 assuming it for n = k. That single link, repeated, reaches every natural number above the base, like toppling a line of dominoes.
Practise Mathematical Induction
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More questions on Mathematical Induction
- Select every statement that is TRUE about mathematical induction.
- An inductive step alone, without a base case, is enough to prove a statement for all n.
- 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.
- An induction proof assumes the very statement it is trying to prove holds for n, then proves it for n+1. Why…
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