How do you prove a statement is true for every positive integer, when there are infinitely many of them to check? You can't test them one at a time, so induction does something cleverer.
Think of an infinite row of dominoes. You show two things: that the first one falls, and that whenever any domino falls it knocks over the next. Those two facts together mean every domino falls, without you ever having to push them individually.
Induction works the same way. Prove the statement for n = 1. Then assume it holds for some n = k, and use that assumption to prove it must also hold for n = k + 1. Since it's true for 1, it must be true for 2; since it's true for 2, it must be true for 3; and so on forever.
Syllabus points covered
- Use the method of mathematical induction to establish a given result
- Apply induction to sums of series, divisibility properties, matrix powers, inequalities and derivatives
- Set out an induction proof with a correct base case, inductive step and conclusion
- Recognise when induction is the appropriate method
Common mistakes
- Omitting the base case, which invalidates the whole proof.
- Assuming the result for n = k + 1 rather than for n = k. You assume it and then prove the next case.
- Not making explicit use of the inductive assumption in the step — if it isn't used, the proof isn't an induction.
- Leaving out the concluding statement about the result holding for all n.
Worksheets
Use these as soon as you've learned the topic and need to practise it. Start with the worksheet, check yourself against the answers, and only then look at the step-by-step solutions.
- Practice worksheet
- Worksheet answers
- Step-by-step solutionsComing soon
Topical past papers
Move on to these once you've worked through the worksheets and are ready for harder, exam-style questions. Real past-paper questions on this topic, with the official mark scheme and full worked solutions.
- Topical questions
- Mark scheme
- Step-by-step worked solutionsComing soon
Looking for whole papers by session? All Further Mathematics past papers