Here is part of a proof that n^2 + n is always even for every integer n. Which reason completes the proof?

n^2 + n = n(n + 1), the product of two consecutive integers, and one of any two consecutive integers is even.

n^2 is always even and n is always even, so their sum is even.

n^2 + n = 2n, which is a multiple of 2.

n^2 + n is a multiple of n, so it is even.

Is algebraic proof on the Foundation paper?

Not in the 'prove algebraically' form. Spec point A6 asks Foundation candidates to show that algebraic expressions are equivalent and to construct arguments, and the extension to proofs is listed for Higher tier only. Foundation students still meet 'show that' questions, but the odd and even number proofs are a Higher tier sight.

How many marks is an algebraic proof question worth on Edexcel Higher?

Usually 3, sometimes 4. A three mark scheme typically gives one mark for correct algebraic forms for the numbers involved, one for a correct expansion of the expression, and one for the simplified result together with a statement that answers the question. A question that asks you to show a statement is false is normally worth 1-2 marks.

How do you write an odd number in algebra?

As 2n + 1, where n is any integer, because 2n is even and one more than an even number is odd. 2n - 1 works equally well. For two consecutive odd numbers use 2n + 1 and 2n + 3; writing n and n + 2 earns nothing, because nothing in that pair says the numbers are odd.

Does one example prove that a statement is true?

No, and the logic here is deliberately one-sided. No number of examples proves a general statement, but a single counter example disproves one outright. So if a question says 'show that Ravi is wrong', give one value that breaks the claim and state what it produces; if it says 'prove', you need a letter standing for any integer.

Algebraic Proof - Edexcel GCSE Higher Maths

Typically 3-4 marks per paper when it appears, usually as one late question · Spec 1MA1

Algebraic proof sits at spec point A6 of Edexcel's 1MA1 specification. Foundation tier is asked to show that two algebraic expressions are equivalent and to construct arguments; the word 'proofs' is added for Higher tier only, which is why a 'prove algebraically that...' question never turns up on a Foundation paper. The whole topic runs on a handful of pieces of notation: $2n$ for an even number, $2n + 1$ for an odd number, $n$ and $n + 1$ for consecutive integers, and $2n + 1$ , $2n + 3$ for consecutive odd ones.

Edexcel normally sets one of these late in a paper for 3-4 marks, and the mark scheme often uses C marks (communication) rather than the M and A marks you meet elsewhere, because what is being credited is the argument rather than the arithmetic. It can appear on any of the three papers; the calculator makes no difference, since every number in the working is a letter. There is a shorter version too, worth 1-2 marks, which asks you to show a statement is false, and there a single counter example finishes the job.

Two habits collect the marks. Bracket the whole of the second expansion before you subtract: written without brackets, $(2n+1)^2 - (2n-1)^2$ collapses to $4n^2 + 4n + 1 - 4n^2 - 4n + 1 = 2$ and the proof is gone. Then finish with a sentence. A final line reading $8n$ does not answer 'prove the difference is always a multiple of 8'; the words 'since $n$ is an integer, $8n$ is a multiple of 8' are what the last mark is for. Practise both halves, the manipulation and the closing statement, because candidates who write neat algebra and stop still lose a third of the question.

Worked example

Prove algebraically that the difference between the squares of any two consecutive odd numbers is always a multiple of 8.

[3 marks]

Let $n$ be an integer. Then $2n - 1$ and $2n + 1$ are two consecutive odd numbers.
C1: a correct algebraic form for an odd number, with $n$ stated to be an integer; $2n+1$ and $2n+3$ would do just as well
The difference between their squares is $(2n+1)^2 - (2n-1)^2 = (4n^2 + 4n + 1) - (4n^2 - 4n + 1)$ .
C1: both squares expanded correctly, with the second expansion kept inside a bracket
This simplifies to $8n$ . Since $n$ is an integer, $8n$ is a multiple of 8, so the difference between the squares of any two consecutive odd numbers is always a multiple of 8.
C1: correct simplification to $8n$ followed by a statement that answers the question in the words it was asked

Practice questions

These are original questions in Edexcel 1MA1 Higher style, and none of them needs a calculator. Because a written proof cannot be marked automatically, most of these questions hand you the setup and ask for the one part that can be checked: the simplified expression, the largest number that is always a factor, the line that completes the argument, or the counter example that kills the statement. Type algebra the way you would write it, for example $8n+8$ ; any equivalent form is accepted.

No account needed: the questions below are always free to check, plus 8 extra questions today. Your score so far updates as you go.

Question 1 Warm-up [1 mark]

$n$ is an integer. Which expression is always an even number?

Answer options for question 1
$2n$ $n + 2$ $2n + 1$ $n^2$
Question 2 Exam pace [3 marks]

Expand and simplify $(2n + 1)^2 - (2n - 1)^2$ .

Your expression (equivalent forms accepted, e.g. 2(x+3) or 2x+6)
Question 3 Stretch [3 marks]

$n$ is an integer. Expand and simplify $(n + 1)^3 - n^3$ .

Your expression (equivalent forms accepted, e.g. 2(x+3) or 2x+6)
Question 4 Stretch [1 mark]

Here is part of a proof that $n^2 + n$ is always even for every integer $n$ . Which reason completes the proof?

Answer options for question 4
$n^2 + n = n(n + 1)$ , the product of two consecutive integers, and one of any two consecutive integers is even. $n^2$ is always even and $n$ is always even, so their sum is even. $n^2 + n = 2n$ , which is a multiple of 2. $n^2 + n$ is a multiple of $n$ , so it is even.
Question 5 Stretch [3 marks]

$n$ is an integer. What is the largest number that $(2n + 3)^2 - (2n - 3)^2$ is always a multiple of?

Your answer (fractions like 1/2 are fine)
Question 6 Stretch [3 marks]

Two consecutive multiples of 5 are $5n$ and $5n + 5$ , where $n$ is an integer. What is the largest number that the difference between their squares is always a multiple of?

Your answer (fractions like 1/2 are fine)

Common mistakes examiners see

Using $n$ and $n + 1$ for an even and an odd number, or $n$ and $n + 2$ for two consecutive odd numbers. Nothing in those expressions forces the numbers to be even or odd, so the proof covers nothing.

Build every odd number from an even one: $2n$ is even, so $2n + 1$ is odd and the next odd number after it is $2n + 3$ . Write the line 'where $n$ is an integer' underneath, since the first mark is for exactly that.
Dropping the bracket when subtracting the second square, so $(2n+1)^2 - (2n-1)^2$ becomes $4n^2 + 4n + 1 - 4n^2 - 4n + 1$ , which simplifies to 2 rather than $8n$ .

Write the subtraction as $(4n^2 + 4n + 1) - (4n^2 - 4n + 1)$ first and change every sign in the second bracket on the next line. The middle mark is for a correct difference, so this one slip costs two marks, not one.
Reaching $8n$ or $6(n+1)$ and stopping. The algebra is right, the question is unanswered, and the final communication mark goes.

Close with a sentence that reuses the wording of the question: 'since $n$ is an integer, $6(n+1)$ is a multiple of 6, so the sum of any three consecutive even numbers is a multiple of 6'.
Testing numbers instead of proving, for example writing ' $3^2 - 1^2 = 8$ and $5^2 - 3^2 = 16$ , so it is always a multiple of 8'. Two cases score nothing on a 'prove algebraically' question.

Examples are only worth marks in the opposite direction. To disprove a statement, one counter example is a complete answer; to prove one, you need a letter standing for any integer.
Claiming the wrong factor at the end, such as reading $6n + 9$ as a multiple of 9 because 9 is the number left behind, or calling $4n + 4$ a multiple of 8 because $4 + 4 = 8$ .

Factorise before you claim anything: $6n + 9 = 3(2n + 3)$ and $4n + 4 = 4(n + 1)$ . Only the factor you can take out of every term is the one you are allowed to state.

Frequently asked questions

Is algebraic proof on the Foundation paper?: Not in the 'prove algebraically' form. Spec point A6 asks Foundation candidates to show that algebraic expressions are equivalent and to construct arguments, and the extension to proofs is listed for Higher tier only. Foundation students still meet 'show that' questions, but the odd and even number proofs are a Higher tier sight.
How many marks is an algebraic proof question worth on Edexcel Higher?: Usually 3, sometimes 4. A three mark scheme typically gives one mark for correct algebraic forms for the numbers involved, one for a correct expansion of the expression, and one for the simplified result together with a statement that answers the question. A question that asks you to show a statement is false is normally worth 1-2 marks.
How do you write an odd number in algebra?: As $2n + 1$ , where $n$ is any integer, because $2n$ is even and one more than an even number is odd. $2n - 1$ works equally well. For two consecutive odd numbers use $2n + 1$ and $2n + 3$ ; writing $n$ and $n + 2$ earns nothing, because nothing in that pair says the numbers are odd.
Does one example prove that a statement is true?: No, and the logic here is deliberately one-sided. No number of examples proves a general statement, but a single counter example disproves one outright. So if a question says 'show that Ravi is wrong', give one value that breaks the claim and state what it produces; if it says 'prove', you need a letter standing for any integer.

Algebraic Proof - Edexcel GCSE Higher Maths

Worked example

Practice questions

Common mistakes examiners see

Frequently asked questions

Related topics