Proofs

[Maximum mark: 5]

1. Prove that $\dfrac{5}{x^2} = \dfrac{5}{x(x-2)}-\dfrac{10}{x^2(x-2)}$. [3]
   

2. Determine the set of numbers $x$ for which the proof in part (a) is valid. [2]
   

[Maximum mark: 4]

Consider two consecutive positive integers, $k$ and $k+1$.

Show that the difference of their squares is equal to the sum of the two integers.

[Maximum mark: 4]

Prove that the sum of three consecutive positive integers is divisible by $3$.

[Maximum mark: 4]

The product of three consecutive integers is increased by the middle integer.

Prove that the result is a perfect cube.