Robinson Tuition
Grade 9 · Higher · free GCSE maths worksheet

Algebraic proof with quadratics and identities

Multi-step proofs.

11 original questions, 41 marks. Print the worksheet, mark it with the answers sheet, and use the solutions to see every step written out by hand.

Worksheet PDFWorked solutions PDFAnswers PDF

The questions

1

Prove algebraically that the difference between the squares of two consecutive integers is equal to the sum of the two integers.

3 marks
Show answer
(n + 1)2 - n2 = 2n + 1 = n + (n + 1)
2

A regular hexagon is drawn inside a circle of radius r so that all six vertices lie on the circle. Each side of the hexagon is equal to r.

The perimeter of the hexagon is less than the circumference of the circle.

Use this to show that π > 3.

3 marks
Show answer
6r 3
3

Prove that x2 − 6x + 11 is greater than 0 for all values of x.

3 marks
Show answer
(x - 3)2 + 2 >= 2 > 0
4

Prove algebraically that the sum of the squares of any three consecutive integers is always 2 more than a multiple of 3.

3 marks
Show answer
(n - 1)2 + n2 + (n + 1)2 = 3n2 + 2
5

Prove algebraically that (3n + 1)2 − (3n − 1)2 is a multiple of 12 for every integer n.

3 marks
Show answer
(3n + 1)2 - (3n - 1)2 = 12n, a multiple of 12
6

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

4 marks
Show answer
(2n + 1)2 - (2n - 1)2 = 8n, a multiple of 8
7

  1. Show that (n + 3)2 − (n − 2)2 ≡ 5(2n + 1)
  2. Hence explain why the difference between (n + 3)2 and (n − 2)2 is always an odd multiple of 5, for every integer n.

4 marks
Show answer
a) both sides equal 10n + 5; b) 2n + 1 is odd, so 5(2n + 1) is 5 times an odd number
8

Prove algebraically that n3 − n is a multiple of 6 for every positive integer n.

4 marks
Show answer
n3 - n = (n - 1)n(n + 1), a product of three consecutive integers, so a multiple of 2 and 3, hence 6
9

Prove algebraically that the square of any odd number is 1 more than a multiple of 8.

4 marks
Show answer
(2n + 1)2 = 4n(n + 1) + 1; n(n + 1) is even, so 4n(n + 1) is a multiple of 8
10

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

6 marks
Show answer
(2a + 1)2 - (2b + 1)2 = 4a(a + 1) - 4b(b + 1) = 8m - 8k, a multiple of 8
11

Prove that the product of four consecutive integers, plus 1, is always a perfect square.

4 marks
Show answer
n(n + 1)(n + 2)(n + 3) + 1 = (n2 + 3n + 1)2

Every question here is original and written by Robinson Tuition. The grade follows the Edexcel grade descriptors and suits AQA and OCR too; this is not an exam-board paper.

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