\(\mathrm { f } ( x ) = \left( 2 + \frac { k x } { 8 } \right) ^ { 7 }\) where \(k\) is a non-zero constant
Find the first 4 terms, in ascending powers of \(x\), of the binomial expansion of \(\mathrm { f } ( x )\). Give each term in simplest form.
Given that, in the binomial expansion of \(\mathrm { f } ( x )\), the coefficients of \(x , x ^ { 2 }\) and \(x ^ { 3 }\) are the first 3 terms of an arithmetic progression,
find, using algebra, the possible values of \(k\).
(Solutions relying entirely on calculator technology are not acceptable.)