Dominoes: Induction is a line of dominoes: knock the first one over, and show each one knocks the next, and they all fall. So a statement is proved for every positive whole number by checking a start and a single step.
The four steps every induction proof shows
- Base case: show the statement is true for the first value, usually (check both sides).
- Assumption: suppose it is true for .
- Inductive step: use that assumption to show it is then true for .
- Conclusion: true for , and each case forcing the next means it holds for all .
Press play to watch the proof build one justified line at a time.
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Don't drop the conclusion: Always finish with the conclusion sentence: name the base case, the \u201ck forces k + 1\u201d link, and \u201ctrue for all \u201d.
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A harder sum: The same four steps work on a harder sum. The trick in the step: to go from to you add the next term to the assumed sum, then factor to reach the target form.
Press play — the key line is factoring (k + 1) out of the bracket.
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Add, then factor: In the step you add the next term to the assumed sum \u2014 never re-sum from the start. The factor always comes out first.
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Induction for divisibility: Induction also proves divisibility. The step students miss: rewrite the expression so the case appears, then use the assumption.
Press play — watch the (8k - 1) from the assumption get pulled out.
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Make the k case appear: The pivot is : split off the assumed multiple, and the leftover ( here) must itself be divisible by the number.
An exam-style question
Use mathematical induction to prove that for all .
Step by step
- Base case: . Put into both sides.
- Assume true for . This is the one line we take as given.
- Inductive step: add to both sides.
- Factor out and tidy the bracket over 4.
- Match the target. That is the formula with .
- Conclusion. True for , and true for forces it true for , so by induction it holds for all .
Final answer
Proved by induction for all .
To get full marks
- Show the base case works: put into both sides and show each equals 1 — writing only “true for ” is not enough.
- Write the assumption as a clear line (“assume true for ”), then add to both sides.
- Reach the form by factoring out and tidying; show this algebra.
- Finish with the full conclusion sentence: the base case, the “ forces ” link, and “for all ”.