Robinson Tuition
Grade 7 · Higher · free GCSE maths worksheet

Algebraic proof

Consecutive numbers, odd and even.

9 original questions, 26 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 sum of an odd number and an even number is always odd.

2 marks
Show answer
2n + (2m + 1) = 2(n + m) + 1, which is odd
2

Sam says, “n2 + n + 11 is a prime number for every positive integer n.”

Show that Sam is wrong.

2 marks
Show answer
n = 10 gives 121 = 11 x 11, which is not prime
3

Prove that n2 + n is even for every integer n.

2 marks
Show answer
n2 + n = n(n + 1), the product of consecutive integers, so it is even
4

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

3 marks
Show answer
(n - 1) + n + (n + 1) = 3n, a multiple of 3
5

Prove algebraically that the sum of the squares of any two consecutive even numbers is a multiple of 4.

3 marks
Show answer
(2n)2 + (2n + 2)2 = 8n2 + 8n + 4 = 4(2n2 + 2n + 1), a multiple of 4
6

Prove algebraically that the product of any two consecutive odd numbers is always 1 less than a multiple of 4.

3 marks
Show answer
(2n - 1)(2n + 1) = 4n2 - 1, which is 1 less than the multiple of 4 given by 4n2
7

Prove algebraically that the sum of the squares of any two consecutive integers is an odd number.

4 marks
Show answer
n2 + (n + 1)2 = 2(n2 + n) + 1, which is odd
8

The nth triangular number is n(n + 1)2.

Prove that the sum of the nth and the (n + 1)th triangular numbers is always a square number.

3 marks
Show answer
n(n+1)/2 + (n+1)(n+2)/2 = (n+1)2
9

The nth triangular number is n(n + 1)2.

Prove that eight times any triangular number, plus 1, is always the square of an odd number.

4 marks
Show answer
8 x n(n+1)/2 + 1 = (2n+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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