A proof explains why a statement is true for every allowed value, not just the ones you tried. Exam questions use “show that”, “prove that” or “explain why”, and the step a calculator cannot supply is the reason. This is a core skill in advanced non-calculator reasoning.
What does a proof need?
A short algebraic proof has three parts:
- Represent the general case with letters, such as an integer n, or consecutive integers n and n + 1.
- Transform the expression with algebra until the structure shows, such as a common factor.
- Justify the last step in words, stating the fact that makes the conclusion true.
Substituting 1, 2 and 3 in place of n can suggest a pattern, but it is not step 3.
Worked example
Prove that (2n + 1)² − 1 is a multiple of 8 for every positive integer n.
Step 1, expand: (2n + 1)² − 1 = 4n² + 4n + 1 − 1 = 4n² + 4n.
Step 2, factorise: 4n² + 4n = 4n(n + 1).
Step 3, justify: n and n + 1 are consecutive integers, so one of them is even. Therefore n(n + 1) is even, which means n(n + 1) = 2k for some integer k.
Step 4, conclude: 4n(n + 1) = 4 × 2k = 8k. So the expression is a multiple of 8.
Check with numbers (not a proof, but a useful test): n = 1 gives 9 − 1 = 8, n = 2 gives 25 − 1 = 24 = 8 × 3, n = 3 gives 49 − 1 = 48 = 8 × 6. All are multiples of 8.
The calculator could produce the last line for chosen values of n. It could not tell you why the expression is always a multiple of 8. The sentence in step 3 is that reason.
The mistake to watch for
A common slip is to test a few values and call it a proof.
Mistaken working: “When n = 1, 2, 3 the answers are 8, 24, 48, which are all multiples of 8, so it is always true.”
The student showed three cases, not all cases.
The correction is to use the letter n, factorise to 4n(n + 1), and give the reason n(n + 1) is even. A claim can also fail after many successes: n² + n + 41 is prime for n = 1 to 39, yet n = 40 gives 1681 = 41².
Check yourself
Try these without a calculator, then open each answer.
1. Show that (n + 3)² − (n − 3)² is a multiple of 12 for every integer n.
Show answer
Expand: (n² + 6n + 9) − (n² − 6n + 9) = 12n. Since n is an integer, 12n is 12 times an integer, so it is a multiple of 12.
2. Give a counterexample to the claim “n² + n + 41 is prime for every positive integer n”.
Show answer
Take n = 40: 1600 + 40 + 41 = 1681, and 41 × 41 = 1681. So 1681 is not prime, and the claim is false.
3. Explain why the sum of any three consecutive integers is divisible by 3.
Show answer
Let the integers be n − 1, n and n + 1. Their sum is 3n, which is 3 times an integer. So it is divisible by 3.
Where this leads next
Next, see checking a result using a second independent method. The non-calculator working trainer and the quadratic structure explorer give you patterns to test before you write a proof.
Some students get the algebra right and lose the mark on the wording. Our teachers read your sentences as closely as your equations in online one-to-one Additional Mathematics tuition.