Algebraic Proof

Show that   is an even number for all positive integer values of .

 is an integer greater than

Prove algebraically that  is always an even number.

Prove that the difference between two consecutive square numbers is always an odd number.
Show clear algebraic working.

 is a multiple of 5

Prove, using algebra, that  is always a multiple of 20

Ali thinks that the value of  will be a prime number for any whole number value of .

Is Ali correct?
You must give a reason for your answer.

is a positive number.

is a negative number.

For each statement, tick the correct box.

	Always true	Sometimes true	Never true
is positive
is positive
is positive
is positive

is an integer.

Prove that    is a square number.

Which of these is a correct identity?

Circle your answer.

Mo says,

   “ will be a prime number for all integer values of  from to ”

Show that Mo is wrong.
You must show that your value of  is not prime.

Tick whether the following statement is true or false.

Give a reason for your answer.

When  is a positive integer, the value of  is always a factor of the value of .

True          False   

Prove that the mean of any four consecutive even integers is an integer.

Bethany says that (2x)² is always greater than or equal to 2x.

Decide whether she is correct or not.
Show your working to justify your decision.

is a positive integer.

Prove that   is a multiple of .

Prove that the difference between two consecutive square numbers is always odd.

Prove that the sum of four consecutive whole numbers is always even.

Give an example to show that the sum of four consecutive integers is not always divisible by 4.

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.

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.

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,