What is important to know about Algebraic proof?

An even number can be written as 2n; an odd number as 2n + 1 (where n is an integer).

What is important to know about Algebraic proof?

Consecutive integers: n, n+1, n+2; consecutive even: 2n, 2n+2; consecutive odd: 2n+1, 2n+3.

What is important to know about Algebraic proof?

To prove an expression is always even, show it equals 2 × (integer).

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Algebra · Higher

Algebraic proof

Algebraic proof uses algebra to prove that a statement is always true (or always false). You show that an expression is equivalent to the required form for all values of the variable.

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Key facts to remember

1An even number can be written as 2n; an odd number as 2n + 1 (where n is an integer).
2Consecutive integers: n, n+1, n+2; consecutive even: 2n, 2n+2; consecutive odd: 2n+1, 2n+3.
3To prove an expression is always even, show it equals 2 × (integer).
4To prove an expression is always odd, show it equals 2 × (integer) + 1.
5Expand and simplify fully, then factorise to show the required property.
6A counterexample (one value that fails) is enough to disprove a statement.

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Worked examples

Example 1

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

Working

Let the integers be n, n + 1, n + 2
Sum = n + (n + 1) + (n + 2) = 3n + 3 = 3(n + 1)
3(n + 1) is always a multiple of 3

AnswerProven: the sum = 3(n + 1) which is always divisible by 3.

Example 2

Prove that (n + 3)² − (n + 1)² is always a multiple of 4.

Working

Expand (n + 3)² = n² + 6n + 9
Expand (n + 1)² = n² + 2n + 1
Difference = (n² + 6n + 9) − (n² + 2n + 1) = 4n + 8 = 4(n + 2)
4(n + 2) is always divisible by 4

AnswerProven: the expression = 4(n + 2), always a multiple of 4.

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Common mistakes

✗Using specific numbers (e.g. n = 3) — this shows an example, not a proof.

✗Not fully expanding brackets before simplifying.

✗Failing to factorise the result to show divisibility.

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Exam tips

✓Always use algebraic expressions (2n, 2n+1 etc.) — never use specific numbers in a proof.

✓End by clearly stating what you have shown, e.g. "This is divisible by 4 for all integer values of n."

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