What does “prove” mean?: To prove something means you show why it is always true.
You cannot just try one example.
You need to use algebra so your answer works for every possible number.
Think of proof like a path: A proof is a short chain of steps:
1. Start with what the question gives you. 2. Change it one step at a time. 3. End with what the question asked you to prove.
Not enough
- Trying one example only
- Starting with the answer
- Writing lines with no explanation
Good proof
- Start from the given information
- Use algebra step by step
- Finish with a sentence explaining why it proves the result
Do not work backwards: If the question says show that, the final answer is already given to you.
Do not start with the final answer.
Start from the expression in the question and work towards the result.
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Name the integer with algebra: Most number proofs start by replacing the number with algebra — then you can add, expand, and factor it.
Every odd number is 2 × (a whole number) + 1 — that's why a single letter, k, stands for them all:
| Odd number | = 2k + 1 | k |
|---|---|---|
| 3 | 2×1 + 1 | 1 |
| 5 | 2×2 + 1 | 2 |
| 7 | 2×3 + 1 | 3 |
| 9 | 2×4 + 1 | 4 |
IB-style question — the sum of two odd numbers is even
Prove that the sum of any two odd numbers is even.
Step by step
- Write each odd number with its OWN letter — use a and b, not k twice, so the two odds can be different (e.g. 3 = 2(1) + 1 and 7 = 2(3) + 1).
- Add them.
- Collect like terms.
- Factor out 2.
Final answer
The result is 2 × (a whole number), so it is even.
The proof pattern — every time:
- Write the number algebraically (2k + 1, 2a + 1, …)
- Add or multiply, then simplify
- Factor out the key number
- Say what the final form means
Know your predicted grade
Take timed mock exams and get detailed feedback on every answer. See exactly where you're losing marks.
Watch a proof build: A proof is just one clear line after another.
Each line must have a reason.
That is what the examiner rewards.
Every line has a reason on the right. Try both proofs.
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Why different letters — not 2k and 2k + 1?: Smart question — and the algebra alone won't warn you. Even = 2k, odd = 2k + 1 gives 2k + (2k + 1) = 4k + 1, which is still odd.
2k and 2k + 1
- They are consecutive — exactly one apart.
- So that only proves it when the odd is one more than the even.
- It covers 4 + 5 and 6 + 7, but never 4 + 9.
2a and 2b + 1
- Two different letters.
- They let the even and odd be any even and any odd.
- That is exactly what “for every” needs.
Factor out the k: To show a number is a multiple of k, manipulate it until you can take out a factor of k — write it as k × (an integer).
IB-style question — a difference of two squares
Prove that (2n + 1)² − (2n − 1)² is a multiple of 8, for every integer n.
Step by step
- It's a difference of two squares — use a² − b² = (a + b)(a − b). Don't expand the brackets.
- Simplify each bracket: the +1 and −1 cancel in one, double up in the other.
- Multiply.
Final answer
8n = 8 × n, which is 8 × an integer, so it is a multiple of 8.
Two moves that save you
- See (…)² − (…)²? Use a² − b² = (a + b)(a − b) — the brackets collapse to a clean multiple, with no expanding.
- To prove a number is never a multiple of k, show it always leaves the same remainder: e.g. 3n² + 2 is 3 × (something) + 2, so it can never be a multiple of 3.
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The method shows up everywhere: Pure proof questions are rare — but the "show that" method is tested in almost every paper, hidden inside other topics: sequences, trig, functions.
The rule never changes: start from what's given, justify each line, reach the target.
IB-style question — the geometric condition
Three consecutive terms of a geometric sequence are a, b and c.
Show that b² = ac.
Step by step
- Geometric means consecutive ratios are equal.
- Cross-multiply.
- Write the result.
Final answer
b² = ac — exactly the "show that" you'll meet in sequence questions.
A given result is a gift: If a "show that" sets up a later part, you may use the given result in that next part — even if you couldn't prove it.
Never let a stuck proof cost you the rest of the question.
The proof checklist: Every "show that / prove" question rewards the same three habits — start right, work deductively, finish cleanly.
IB-style question — a difference of two squares
Prove that (n + 3)² − (n − 3)² is a multiple of 12, for every positive integer n.
Step by step
- It's a difference of two squares — use a² − b² = (a + b)(a − b). Don't expand the brackets.
- Simplify each bracket: the +3 and −3 cancel in one, double up in the other.
- Multiply.
Final answer
12n = 12 × n, which is 12 × an integer, so it is a multiple of 12.
Do
- Start from what's given, or one side only.
- Name integers with algebra (2k, 2k + 1).
- Give a reason for each line; factor out the k.
- Finish at the target and conclude in words.
Don't
- Start by writing the result (working backwards).
- Prove by example — a few cases is not "for all".
- Reuse one letter for two independent unknowns.
- Stop one line early without the conclusion.