- (i) (a) Find, in ascending powers of \(x\), the 2nd, 3rd and 5th terms of the binomial expansion of
$$( 3 + 2 x ) ^ { 6 }$$
For a particular value of \(x\), these three terms form consecutive terms in a geometric series.
(b) Find this value of \(x\).
(ii) In a different geometric series,
- the first term is \(\sin ^ { 2 } \theta\)
- the common ratio is \(2 \cos \theta\)
- the sum to infinity is \(\frac { 8 } { 5 }\)
(a) Show that
$$5 \cos ^ { 2 } \theta - 16 \cos \theta + 3 = 0$$
(b) Hence find the exact value of the 2nd term in the series.