1. 1
   A maths club investigates proofs about integers. They study consecutive integers $n$, $n+1$, $n+2$; even numbers written as $2n$; and odd numbers written as $2n+1$.

   (a)

   Which expression correctly represents 'the sum of three consecutive integers, the smallest of which is $n$', written before simplifying?

   [1 mark]

   • A$3n+2$
   • B$n^3$
   • C$3n$
   • D$n+(n+1)+(n+2)$

   (b)

   Simplifying $n+(n+1)+(n+2)$, which expression shows the sum of three consecutive integers is always a multiple of 3?

   [1 mark]

   • A$3(n+1)$
   • B$3n+2$
   • C$3n+1$
   • D$n^3+3$

   (c)

   If $2n$ represents any even number, which expression represents the square of an even number?

   [1 mark]

   • A$4n$
   • B$2n^2$
   • C$4n^2$
   • D$2n+2$

   Total for question 1: 3 marks

2. 2
   A student is proving results about odd and even numbers using algebra. The odd number is written as $2n+1$ for integer $n$.

   (a)

   Prove algebraically that the square of any odd number is always odd. Let the odd number be $2n+1$, where $n$ is an integer.

   [2 marks]

   (b)

   Prove algebraically that the sum of two odd numbers is always even. Let the two odd numbers be $2m+1$ and $2n+1$, where $m$ and $n$ are integers.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A teacher sets a reasoning task on properties of consecutive integers $n$ and $n+1$, and on the difference of two squares of consecutive integers.

   (a)

   Prove that the product of two consecutive integers is always even. Let the integers be $n$ and $n+1$, where $n$ is an integer.

   [3 marks]

   (b)

   Show that the difference between the squares of two consecutive integers $n+1$ and $n$ is always equal to the sum of the two integers.

   [3 marks]

   Total for question 3: 6 marks

4. 4
   A mathematics olympiad problem set asks students to prove several results about consecutive even numbers $2n$ and $2n+2$, and about a general algebraic identity involving three consecutive integers $n-1$, $n$, $n+1$.

   (a)

   Prove algebraically that the sum of the squares of any two consecutive even numbers is always a multiple of 4. Let the two consecutive even numbers be $2n$ and $2n+2$, where $n$ is an integer.

   [4 marks]

   (b)

   Show that the product of three consecutive integers $(n-1)$, $n$ and $(n+1)$ can be written as $n^3-n$.

   [4 marks]

   (c)

   Using the result that the product of three consecutive integers $(n-1)$, $n$ and $(n+1)$ equals $n^3-n$, prove that this product is always divisible by 6 for any integer $n \geq 2$, by explaining why it must be divisible by both 2 and 3.

   [5 marks]

   Total for question 4: 13 marks

End of questions