Algebraic Proof (Edexcel 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 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

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.

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

$n$ 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.

$a^2–b^2\equiv (a+b)(a–b)$

• $a$ and $b$ are positive whole numbers with $a>b$

• $a^2–b^2$ is a prime number.

Why are $a$ and $b$ consecutive numbers?

The $n$th term for the sequence of even numbers, $P$, is $2n$.

The $n$th term for the sequence of odd numbers, $Q$, is $2n-1$.

A new sequence $R$ is formed using the two sequences. Each term of $R$ is calculated by squaring the corresponding term of $P$ and then subtracting the corresponding term of $Q$.

Show that all the terms in the sequence $R$ are odd.

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.

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.

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, $L–T=99.$