Algebraic Proof& Identities.

Q: Why can you not prove something by substituting numbers?

Substituting numbers only verifies the statement for those specific values. It does not show the statement is true for ALL values. An algebraic proof uses general variables (like n) to cover every possible case simultaneously. In IGCSE exams, testing with examples scores zero marks on proof questions.

Q: What algebraic representations do you need to know for IGCSE proof?

The key representations are: any integer = n, even number = 2n, odd number = 2n + 1, consecutive integers = n, n + 1, n + 2, consecutive even numbers = 2n, 2n + 2, consecutive odd numbers = 2n + 1, 2n + 3, and a multiple of k = k multiplied by an expression in brackets. These form the building blocks of every proof question.

Q: How do you prove an identity in IGCSE Maths?

Start with one side of the identity (usually the left-hand side). Expand, simplify and manipulate it algebraically until it matches the other side. Write each step clearly and finish with a conclusion such as "LHS = RHS, therefore the identity is proven." Never work on both sides at the same time.

Q: Is algebraic proof tested on both IGCSE tiers?

Algebraic proof is a Higher tier only topic, typically appearing as a 3-5 mark question on Paper 1 or Paper 2. It usually appears towards the end of the paper as a grade 8 or 9 question. Foundation tier students are not expected to write algebraic proofs.

Higher tier essential: use general variables to prove statements are always true, not just sometimes.

Hire a Tutor

6+ Years

Full-Time Professional Educators

In-Person Only

2,000+ Families

What is algebraic proof and how do you prove identities in IGCSE Maths?

Algebraic proof uses general variables (like n for any integer, 2n for even, 2n + 1 for odd) to show a mathematical statement is true for ALL cases, not just specific examples. You expand, simplify and factorise to reach the required conclusion. Proving an identity means starting with one side of an equation (usually the LHS) and manipulating it algebraically until it equals the other side. This is a Higher tier topic worth 3–5 marks.

Tier: HigherTopic: AlgebraPapers: 1 & 2

Source: Edexcel IGCSE Mathematics (9-1) Specification 4MA1

Key Takeaway

Testing with examples is NOT proof — you must use algebra with general variables. Learn the dictionary: n = integer, 2n = even, 2n + 1 = odd. The proof is complete only when you write a concluding sentence linking your algebra to the original statement. A missing conclusion loses the final mark every time.

What is algebraic proof and why can't you use examples?

There is an important distinction between verification and proof. Verification checks that a statement works for particular numbers. Proof shows it must work for every possible case.

For example, you could verify that "the sum of two odd numbers is even" by testing 3 + 5 = 8. But this only confirms one pair. A proof uses algebra to show it for all odd numbers: (2m + 1) + (2n + 1) = 2m + 2n + 2 = 2(m + n + 1), which is even because it has a factor of 2.

In IGCSE exams, testing specific numbers scores zero marks on proof questions. You must use general variables throughout and arrive at the conclusion algebraically.

What is the dictionary of proof?

Before writing any proof, you need the correct algebraic representation for each type of number. Think of this as your "proof dictionary":

Any integer: n
Even number: 2n
Odd number: 2n + 1
Consecutive integers: n, n + 1, n + 2
Consecutive even numbers: 2n, 2n + 2
Consecutive odd numbers: 2n + 1, 2n + 3
Multiple of k: k(…) — show a factor of k in the expression

Use different letters (m, n, p) when you need independent numbers. For instance, "any two even numbers" should be 2m and 2n (not 2n and 2n, which would be the same even number twice). Consecutive numbers use the same variable with increments.

How do you prove identities?

An identity is an equation that is true for all values of the variable (written with the ≡ symbol). To prove one, use the LHS = RHS method:

Step 1: Start with the left-hand side (LHS) of the identity.
Step 2: Expand brackets, simplify terms, and factorise as needed.
Step 3: Show that your simplified expression is identical to the right-hand side (RHS).
Step 4: Write a conclusion: "LHS = RHS, therefore the identity is proven."

The key rule is: work on one side only. Do not manipulate both sides simultaneously, as this assumes the result you are trying to prove. Start from LHS and arrive at RHS (or vice versa).

Worked Examples

Grade 8

Prove that the sum of 3 consecutive integers is always a multiple of 3.

Step 1 — Define: Let the three consecutive integers be n, n + 1 and n + 2.

Step 2 — Add them: n + (n + 1) + (n + 2) = 3n + 3

Step 3 — Factorise: 3n + 3 = 3(n + 1)

Step 4 — Conclude: 3(n + 1) has a factor of 3, so the sum is always a multiple of 3. &marker;

The concluding sentence is essential for the final mark.

Grade 9

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

Step 1 — Define: Let the two consecutive odd numbers be (2n + 1) and (2n + 3).

Step 2 — Square both: (2n + 3)² = 4n² + 12n + 9 and (2n + 1)² = 4n² + 4n + 1

Step 3 — Subtract: (2n + 3)² − (2n + 1)² = 4n² + 12n + 9 − 4n² − 4n − 1 = 8n + 8

Step 4 — Factorise: 8n + 8 = 8(n + 1)

Step 5 — Conclude: 8(n + 1) has a factor of 8, so the difference is always a multiple of 8. &marker;

Notice how the 4n² terms cancel perfectly, leaving only linear terms that factorise neatly.

Grade 9

Prove that x² + 6x + 10 is always positive for all real values of x.

Step 1 — Complete the square: x² + 6x + 10 = (x + 3)² − 9 + 10 = (x + 3)² + 1

Step 2 — Reason about the expression: (x + 3)² ≥ 0 for all real x (a square is never negative).

Step 3 — Add the constant: Therefore (x + 3)² + 1 ≥ 0 + 1 = 1.

Step 4 — Conclude: The minimum value of x² + 6x + 10 is 1, which is greater than 0. Therefore the expression is always positive. &marker;

Completing the square is a powerful proof technique for "always positive" questions.

Common Mistakes

Substituting numbers instead of using algebra: Writing "I tried 3, 5 and 7 and it worked" is verification, not proof. You will score zero marks. Always use general variables.

Missing the concluding sentence: The final mark on a proof question is for the conclusion. You must link your algebra to the original statement, e.g. "3(n + 1) has factor 3, so it is a multiple of 3."

Incorrect definitions: Using n to represent an odd number is wrong — n is any integer. An odd number must be written as 2n + 1. Similarly, using n and n + 1 for consecutive even numbers is incorrect; use 2n and 2n + 2.

Consecutive odd numbers: These are 2n + 1 and 2n + 3 (gap of 2), not 2n + 1 and 2n + 2. The number 2n + 2 is even, which would invalidate the proof.

Exam Tips

Always write a concluding sentence: Link your final algebraic expression to the question. State clearly why the factor or form proves the statement. This is worth 1 mark and is frequently missed.

Show the key factor clearly: If proving "multiple of 8", your final line should show 8(…) explicitly. If proving "always positive", show the squared term plus a positive constant.

Use correct algebraic representations from the start: Getting the definitions right in Step 1 determines whether the entire proof works. Double-check even vs odd, consecutive vs independent.

Practise expanding brackets accurately: Most proof questions hinge on correct expansion of (2n + 1)² or similar. One sign error cascades through the whole proof. Write out every term.

Algebra Hub

All algebra topics in one place

Worksheets & Answers

Free algebra practice papers

Formula Sheet

Visual guide to key formulae

IGCSE Tutors in Dubai

Specialist in-home IGCSE support

Maths Tutors in Dubai

Expert maths tutoring at home

Contact GetYourTutors — IGCSE Maths Algebraic Proof

Phone: (+971) 4-313-2715 | Mobile: 050-947-9432

WhatsApp: 050-947-9432

Email: info@getyourtutors.com

Emirates Towers, Office Tower, Level 41, Sheikh Zayed Road, PO Box 31003, Dubai, UAE

Last updated: March 2026

Frequently Asked Questions

Everything you need to know about our private tutoring services in Dubai.

Why can you not prove something by substituting numbers?

What algebraic representations do you need to know for IGCSE proof?

How do you prove an identity in IGCSE Maths?

Is algebraic proof tested on both IGCSE tiers?

Need Help With IGCSE Higher Tier Proof?

A dedicated maths tutor builds the algebraic reasoning skills needed for top grades. In-home across Dubai.

Hire a Tutor WhatsApp Us

Loading…

What is algebraic proof and why can't you use examples?

There is an important distinction between verification and proof. Verification checks that a statement works for particular numbers. Proof shows it must work for every possible case.

In IGCSE exams, testing specific numbers scores zero marks on proof questions. You must use general variables throughout and arrive at the conclusion algebraically.

What is the dictionary of proof?

Before writing any proof, you need the correct algebraic representation for each type of number. Think of this as your "proof dictionary":

Any integer: n
Even number: 2n
Odd number: 2n + 1
Consecutive integers: n, n + 1, n + 2
Consecutive even numbers: 2n, 2n + 2
Consecutive odd numbers: 2n + 1, 2n + 3
Multiple of k: k(…) — show a factor of k in the expression

How do you prove identities?

An identity is an equation that is true for all values of the variable (written with the ≡ symbol). To prove one, use the LHS = RHS method:

Step 1: Start with the left-hand side (LHS) of the identity.
Step 2: Expand brackets, simplify terms, and factorise as needed.
Step 3: Show that your simplified expression is identical to the right-hand side (RHS).
Step 4: Write a conclusion: "LHS = RHS, therefore the identity is proven."

The key rule is: work on one side only. Do not manipulate both sides simultaneously, as this assumes the result you are trying to prove. Start from LHS and arrive at RHS (or vice versa).

Worked Examples

Grade 8

Prove that the sum of 3 consecutive integers is always a multiple of 3.

Step 1 — Define: Let the three consecutive integers be n, n + 1 and n + 2.

Step 2 — Add them: n + (n + 1) + (n + 2) = 3n + 3

Step 3 — Factorise: 3n + 3 = 3(n + 1)

Step 4 — Conclude: 3(n + 1) has a factor of 3, so the sum is always a multiple of 3. &marker;

The concluding sentence is essential for the final mark.

Grade 9

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

Step 1 — Define: Let the two consecutive odd numbers be (2n + 1) and (2n + 3).

Step 2 — Square both: (2n + 3)² = 4n² + 12n + 9 and (2n + 1)² = 4n² + 4n + 1

Step 3 — Subtract: (2n + 3)² − (2n + 1)² = 4n² + 12n + 9 − 4n² − 4n − 1 = 8n + 8

Step 4 — Factorise: 8n + 8 = 8(n + 1)

Step 5 — Conclude: 8(n + 1) has a factor of 8, so the difference is always a multiple of 8. &marker;

Notice how the 4n² terms cancel perfectly, leaving only linear terms that factorise neatly.

Grade 9

Prove that x² + 6x + 10 is always positive for all real values of x.

Step 1 — Complete the square: x² + 6x + 10 = (x + 3)² − 9 + 10 = (x + 3)² + 1

Step 2 — Reason about the expression: (x + 3)² ≥ 0 for all real x (a square is never negative).

Step 3 — Add the constant: Therefore (x + 3)² + 1 ≥ 0 + 1 = 1.

Step 4 — Conclude: The minimum value of x² + 6x + 10 is 1, which is greater than 0. Therefore the expression is always positive. &marker;

Completing the square is a powerful proof technique for "always positive" questions.

Common Mistakes

Substituting numbers instead of using algebra: Writing "I tried 3, 5 and 7 and it worked" is verification, not proof. You will score zero marks. Always use general variables.

Consecutive odd numbers: These are 2n + 1 and 2n + 3 (gap of 2), not 2n + 1 and 2n + 2. The number 2n + 2 is even, which would invalidate the proof.

Exam Tips

Always write a concluding sentence: Link your final algebraic expression to the question. State clearly why the factor or form proves the statement. This is worth 1 mark and is frequently missed.

Show the key factor clearly: If proving "multiple of 8", your final line should show 8(…) explicitly. If proving "always positive", show the squared term plus a positive constant.

Use correct algebraic representations from the start: Getting the definitions right in Step 1 determines whether the entire proof works. Double-check even vs odd, consecutive vs independent.

Practise expanding brackets accurately: Most proof questions hinge on correct expansion of (2n + 1)² or similar. One sign error cascades through the whole proof. Write out every term.