5 The first three terms of an arithmetic progression are \(4 , x\) and \(y\) respectively. The first three terms of a geometric progression are \(x , y\) and 18 respectively. It is given that both \(x\) and \(y\) are positive.
- Find the value of \(x\) and the value of \(y\).
- Find the fourth term of each progression.
\includegraphics[max width=\textwidth, alt={}, center]{2d5f452d-f820-40fc-9e22-9d3ac4f0698b-08_389_716_260_712}
The diagram shows a triangle \(A B C\) in which \(B C = 20 \mathrm {~cm}\) and angle \(A B C = 90 ^ { \circ }\). The perpendicular from \(B\) to \(A C\) meets \(A C\) at \(D\) and \(A D = 9 \mathrm {~cm}\). Angle \(B C A = \theta ^ { \circ }\). - By expressing the length of \(B D\) in terms of \(\theta\) in each of the triangles \(A B D\) and \(D B C\), show that \(20 \sin ^ { 2 } \theta = 9 \cos \theta\).
- Hence, showing all necessary working, calculate \(\theta\).