Algebraic Proof (OCR GCSE Maths: Higher): Exam Questions

Prove that

${\left(2n+3\right)}^2-{\left(2n-3\right)}^2$  is a multiple of $8$ 

for all positive integer values of $n$.

Prove that the square of an odd number is always 1 more than a multiple of 4

Prove that , for all positive values of $n$,

$\frac{{\left(n+2\right)}^2-{\left(n+1\right)}^2}{2n^2+3n}=\frac{1}{n}$

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

Prove that  (2x + 1)(3x + 2) + x (3x + 5) + 2  is a perfect square.

Gemma says

   The equation (2x + 1)(3x + 2) + x (3x + 5) + 2 = -12 has no solutions.

Explain Gemma’s reasoning.

n is an integer.

Explain why 2n + 1 is an odd number.

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

Use algebra to prove that the sum of the squares of any two consecutive integers is always odd.

The lengths of the sides of a right-angled triangle are all integers. Prove that if the lengths of the two shortest sides are even, then the length of the third side must also be even.

Express as a single fraction.

$\frac{m+1}{n+1}-\frac{m}{n}$

Simplify your answer.

Using your answer to part (a), prove that if m and n are positive integers and $m<n$, then

$\frac{m+1}{n+1}-\frac{m}{n}>0$

Prove algebraically that the product of any two odd numbers is always an odd number.

Show that $x(x-1)\left(x+1\right)=x^3-x$

Prove that the difference between a whole number and the cube of this number is always a multiple of $6$.

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

$n$ is the middle integer of three consecutive positive integers.

The three integers are multiplied to give a product.

$n$ is then added to the product.

Prove that the result is a cube number.

Expressions for consecutive triangular numbers are

$\frac{n\left(n+1\right)}{2}$ and $\frac{\left(n+1\right)\left(n+2\right)}{2}$

Prove that the sum of two consecutive triangular numbers is always a square number.

$n$ is a positive integer.

Prove algebraically that  $2n^2\left(\frac{3}{n}+n\right)+6n\left(n^2-1\right)$ is a cube number.

$c$ is a positive integer.

Prove that  $\frac{6c^3+30c}{3c^2+15}$  is an even number.

i) Factorise         $2t^2+5t+2$

[2]

ii) $t$ is a positive whole number.

The expression  $2t^2+5t+2$ can never have a value that is a prime number.

Explain why.

[1]

$n$ is an integer.

Prove algebraically that the sum of  $\frac{1}{2}n(n+1)$ and $\frac{1}{2}(n+1)(n+2)$ is always a square number.

Here are the first five terms of an arithmetic sequence.

7      13      19      25      31 

Prove that the difference between the squares of any two terms of the sequence is always a multiple of 24.

Given that $n$ can be any integer such that $n>1$, prove that $n^2-n$ is never an odd number.

The product of two consecutive positive integers is added to the larger of the two integers.

Prove that the result is always a square number.

The diagram shows a cross placed on a number grid.

$L$ is the product of the left and right numbers of the cross.
$T$ is the product of the top and bottom numbers of the cross.
$M$ is the middle number of the cross.

Show that when $M=35,L–T=99.$

Prove that, for any position of the cross on the number grid above, $L–T=99.$

Prove that when the sum of the squares of any two consecutive odd numbers is divided by 8, the remainder is always 2
Show clear algebraic working.

Using algebra, prove that, given any 3 consecutive whole numbers, the sum of the square of the smallest number and the square of the largest number is always 2 more than twice the square of the middle number.

Using algebra, prove that, given any 3 consecutive even numbers, the difference between the square of the largest number and the square of the smallest number is always 8 times the middle number.

Here are the first four terms of a sequence of fractions.

$\frac{1}{1}\frac{2}{3}\frac{3}{5}\frac{4}{7}$

The numerators of the fractions form the sequence of whole numbers $1234...$
The denominators of the fractions form the sequence of odd numbers $1357...$

Write down an expression, in terms of $n$, for the $n\mathrm{th}$ term of this sequence of fractions.

Using algebra, prove that when the square of any odd number is divided by $4$ the remainder is $1$.

The table gives information about the first six terms of a sequence of numbers.

Prove algebraically that the sum of any two consecutive terms of this sequence is always a square number.

Prove that $x^2+x+1$ is always positive.