- (a) Show that the binomial expansion of
$$( 4 + 5 x ) ^ { \frac { 1 } { 2 } }$$
in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\) is
$$2 + \frac { 5 } { 4 } x + k x ^ { 2 }$$
giving the value of the constant \(k\) as a simplified fraction.
(b) (i) Use the expansion from part (a), with \(x = \frac { 1 } { 10 }\), to find an approximate value for \(\sqrt { 2 }\) Give your answer in the form \(\frac { p } { q }\) where \(p\) and \(q\) are integers.
(ii) Explain why substituting \(x = \frac { 1 } { 10 }\) into this binomial expansion leads to a valid approximation.