1.4 Proof & Reasoning

Prove that  is positive for all values of .    

Prove that .

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

Prove that  for all values of

 Prove that the square of an even number is a multiple of 4.

Factorise .

Hence show that 

Given that  is even, write down whether and  are odd or even.

Hence deduce whether  is odd or even. Justify your answer.

Show that  = , where 

Hence, or otherwise, prove that is a multiple of 8.

Prove that the product of two odd numbers is odd.

The sum of squares of two consecutive integers is 313.  Find the possible values of the integers.

Prove that the sum of the cubes of any two consecutive odd integers is divisible by four.

Prove that 

State any values of  for which this mathematical statement does not hold true.

Prove that there are no integers and  that satisfy the equation

Prove the binomial coefficient identity 

Prove that the sum of all integers between 600 and 1400 (inclusive) that are not divisible by 7 is equal to