Try a short problem first. Then use the topic map to plan what to learn next.

Jump to the full topic map · Continue the four-subject practice plan

Worked example: why the first n odd numbers sum to n²

For every positive integer n, prove that 1 + 3 + 5 + ⋯ + (2n − 1) = n². Checking n = 1, 2 and 3 suggests a pattern, but does not prove it for every positive integer.

Base case. For n = 1, the left side is 1 = 1².

Inductive step. Assume the sum through 2k − 1 equals k² for some positive integer k. The next odd number is 2k + 1. Adding it gives k² + 2k + 1 = (k + 1)². Thus truth for k implies truth for k + 1.

The base case and the inductive step prove the statement for all positive integers. Notice that the assumption concerned k; we did not assume the result for k + 1.

Practice: find the missing reason

  1. A student says “n² is even, so n is even”. Prove the implication.
  2. Prove that the sum of two rational numbers is rational.
Solutions

1. If n were odd, n = 2k + 1 for an integer k. Then n² = 2(2k² + 2k) + 1 would be odd, contradicting the assumption. Therefore n is even. State that n is an integer; parity is an integer property.

2. Write the numbers as a/b and c/d with integers a,b,c,d and b,d nonzero. Their sum is (ad + bc)/(bd). The numerator and denominator are integers, and bd is nonzero. This is exactly the definition of a rational number.

These are teaching examples written for this guide, not past-paper questions or an official marking scheme.

Topic map and learning goals

This strand should run through all four subjects from the beginning.

  1. Direct proof: start from assumptions and establish a chain of justified steps.
  2. Contrapositive: prove the logically equivalent opposite implication when it is simpler.
  3. Contradiction: identify exactly which assumption becomes impossible.
  4. Induction: state the proposition, base case and inductive implication explicitly; add strong induction when needed.
  5. Case division: make the cases exhaustive and avoid counting overlaps twice.
  6. Construction: give an object or strategy and verify every condition.
  7. Bounding: establish an attainable bound rather than guessing an optimum.
  8. Descent and extremal arguments: choose a measure that really decreases, or a legitimate minimal object.
  9. Working backwards: use the target to discover a route, then present a valid forward proof.
  10. Solution checking: test assumptions, domains, equality cases, edge cases and converses.

Useful habits include drawing a diagram, considering small cases, identifying symmetry, proving a lemma, and rewriting a failed attempt to explain what was missing. Full solutions need coherent prose as well as equations.