Algebraic Proof (OCR GCSE Maths)

Prove that

 is a multiple of  

for all positive integer values of .

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

Prove that , for all positive values of ,

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.

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.

Simplify your answer.

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

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

Show that 

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

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

is the middle integer of three consecutive positive integers.

The three integers are multiplied to give a product.

is then added to the product.

Prove that the result is a cube number.

Expressions for consecutive triangular numbers are

and 

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

 is a positive integer.

Prove algebraically that   is a cube number.

is a positive integer.

Prove that    is an even number.

is an integer.

Prove algebraically that the sum of  and  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 can be any integer such that , prove that  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.

is the product of the left and right numbers of the cross.
is the product of the top and bottom numbers of the cross.
is the middle number of the cross.

Show that when 

Prove that, for any position of the cross on the number grid above, 

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.

The numerators of the fractions form the sequence of whole numbers 
The denominators of the fractions form the sequence of odd numbers 

Write down an expression, in terms of , for the  term of this sequence of fractions.

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

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

Term number	1	2	3	4	5	6
Term of sequence

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

Prove that is always positive.