12. Consider the binomial expansion of \(\left( 1 + \frac { x } { 5 } \right) ^ { n }\) in ascending powers of \(x\), where \(n\) is a positive integer.
i. Write down the first four terms of the expansion, giving the coefficients as polynomials in \(n\). The coefficients of the second, third and fourth terms of the expansion are consecutive terms of an arithmetic sequence.
ii. Show that \(n ^ { 3 } - 33 n ^ { 2 } + 182 n = 0\).
iii. Hence find the possible values of \(n\) and the corresponding values of the common difference.