Explain why \(n ( n + 1 )\) is a multiple of 2 when \(n\) is an integer.
Given that
$$\mathrm { f } ( n ) = n \left( n ^ { 2 } + 5 \right)$$
show that \(\mathrm { f } ( k + 1 ) - \mathrm { f } ( k )\), where \(k\) is a positive integer, is a multiple of 6 .
Prove by induction that \(\mathrm { f } ( n )\) is a multiple of 6 for all integers \(n \geqslant 1\).