Key Idea: A proof shows a statement is always true, not just true for one example. You start from what's given, justify every line with algebra, and finish with the target. It's a Paper 1, by-hand skill — and the show that method is hidden inside sequence, trig and function questions on both papers.
🔢 Name the numbers with algebra
- any integer — one letter stands for every case
- even: a multiple of 2
- odd: one more than an even number
Two independent odds need two letters: 2a + 1 and 2b + 1. Two consecutive numbers share one: n and n + 1 — they're locked one apart. Pick the tidiest layout (n − 1, n, n + 1 when there's a middle term).
✖️ Show a multiple — factor it out
| Goal | Method | Conclusion line |
|---|---|---|
| Prove it is a multiple of k | Manipulate until you can take out a factor of k. | k(…) = k × an integer, so it's a multiple of k. |
| Prove it's never a multiple of k | Show it always leaves the same remainder — write it as k(…) + r. | Always k × integer + r, so never divisible by k. |
| See (…)² − (…)² | Use a² − b² = (a + b)(a − b) — don't expand the brackets. | The brackets collapse straight to a clean multiple. |
≡ Identities: transform one side
| Equation (=) | Identity (≡) | |
|---|---|---|
| True for… | some values | every value |
| You… | solve it | prove it |
| Method | rearrange both sides | start on one side, turn it into the other |
Tip: Start with the messier side. Polynomial → expand every bracket, then collect like terms. Rational → put over a common denominator, combine, then simplify. Never move terms across the ≡.
✏️ IB-style worked examples
IB-style question — prove a sum is even
Prove that the sum of any two odd numbers is even.
Step by step:
Two independent odds need two letters.
Add and collect like terms.
Factor out 2.
2(a + b + 1) is 2 × an integer, so the sum is even. ∎
IB-style question — prove a multiple of 3
Prove that the sum of any three consecutive integers is a multiple of 3.
Step by step:
Write three consecutive integers.
Add and collect like terms.
Take out a factor of 3.
3(n + 1) is 3 × an integer, so the sum is a multiple of 3. ∎
IB-style question — prove an identity
Prove the identity (x + 4)(x − 1) ≡ x² + 3x − 4.
Step by step:
Start with the left side (the brackets) and multiply out.
Collect like terms.
This is the right-hand side, so (x + 4)(x − 1) ≡ x² + 3x − 4. ∎
Important: One example is not a proof — show that means prove it for every case, using algebra. And when the result is given, don't start from the answer. Begin with the expression you're given (or one side of an identity) and work towards the target, with a reason on every line.
Tap each card to reveal the answer.
How do you write a general even number? An odd one? Even = 2k, odd = 2k + 1, where k is any integer.
Write three consecutive integers in algebra n, n + 1, n + 2 (or n − 1, n, n + 1 — pick the tidiest).
To prove a result is a multiple of 8, what's the final line? Get it to 8 × (an integer) — e.g. 8n = 8 × n, so it's a multiple of 8.
What's the fast move for (2n + 1)² − (2n − 1)²? Difference of squares: a² − b² = (a + b)(a − b) — the brackets collapse, no expanding.
= or ≡ — which do you solve, which do you prove? = is an equation, true for some values → solve. ≡ is an identity, true for all → prove.
How do you prove an identity with fractions? Take one side, put over a common denominator, combine, then simplify to the other side.
Exam Tips
- Replace the number with algebra first: even = 2k, odd = 2k + 1, consecutive = n, n + 1, n + 2.
- Use a separate letter for each independent unknown; share one letter only for consecutive values.
- Multiple of k? Factor out k. Never a multiple? Show a constant remainder: k(…) + r.
- Spot (…)² − (…)² → use a² − b² = (a + b)(a − b) instead of expanding.
- Identity: start on the messier side and transform it into the other — never substitute one value.
- Finish with a sentence: state what you've shown. The conclusion line earns a mark.