8. $$\mathrm { f } ( x ) = \left( 1 + \frac { x } { k } \right) ^ { n } , \quad k , n \in \mathbb { N } , \quad n > 2 .$$ Given that the coefficient of \(x ^ { 3 }\) is twice the coefficient of \(x ^ { 2 }\) in the binomial expansion of \(\mathrm { f } ( x )\),

1. prove that \(n = 6 k + 2\). Given also that the coefficients of \(x ^ { 4 }\) and \(x ^ { 5 }\) are equal and non-zero,
2. form another equation in \(n\) and \(k\) and hence show that \(k = 2\) and \(n = 14\). Using these values of \(k\) and \(n\),
3. expand \(\mathrm { f } ( x )\) in ascending powers of \(x\), up to and including the term in \(x ^ { 5 }\). Give each coefficient as an exact fraction in its lowest terms