- Ch. 1The First Principle of InductionKnock over the first domino and make each one topple the next. All of them fall.
- Ch. 2The Second Principle of InductionTo prove P(k+1), use all earlier cases, not just P(k). You may need more bases.
- Ch. 3Well-Ordering and Infinite DescentPositive integers cannot shrink forever, so no solution can make a smaller one.
- Ch. 4General Terms and SumsSplit each term as “this minus next”; the middle cancels, only the ends survive.
- Ch. 5Arithmetic and Geometric SequencesArithmetic: add it in reverse. Geometric: multiply by q, shift and subtract.
- Ch. 6Higher-Order Arithmetic Sequences and DifferencesSubtract neighbors until a row is constant, then add back up to extend the list.
- Ch. 7Recursive SequencesLinear recurrences: solve the characteristic equation, fit the initial values.
- Ch. 8Periodic SequencesWhen finitely many states decide the next term, a state repeats and it cycles.
- Ch. 9The Fibonacci SequenceDomino tilings count Fibonacci numbers; one cut gives its addition formula.
- Ch. 10Several Proofs of the AM–GM InequalityStuck at n ⇒ n+1? Double 2, 4, 8, …, then fill back down: backward induction.
- Ch. 11Choosing the Right Step SizeEach domino knocks over the k-th next: a big step, but k base cases to check.
- Ch. 12Choosing What to Induct OnSeveral variables? See which smaller cases the step needs, then choose.
- Ch. 13Changing the StatementStuck? Prove more: a stronger claim gives a stronger hypothesis for n → n+1.
- Ch. 14Guess, Then ProveWork out small cases, guess a formula, then nail it down by induction.
- Ch. 15Existence Problems for SequencesShow existence by building it term by term, or show that “none” is absurd.