7. A geometric series is \(a + a r + a r ^ { 2 } + \ldots\)
- Prove that the sum of the first \(n\) terms of this series is given by \(S _ { n } = \frac { a \left( 1 - r ^ { n } \right) } { 1 - r }\).
The second and fourth terms of the series are 3 and 1.08 respectively.
Given that all terms in the series are positive, find - the value of \(r\) and the value of \(a\),
- the sum to infinity of the series.
\includegraphics[max width=\textwidth, alt={}, center]{1033051d-18bf-4734-a556-4c8e1c789992-4_764_1159_294_299}
Fig. 2 shows part of the curve with equation \(y = x ^ { 3 } - 6 x ^ { 2 } + 9 x\). The curve touches the \(x\)-axis at \(A\) and has a maximum turning point at \(B\). - Show that the equation of the curve may be written as \(y = x ( x - 3 ) ^ { 2 }\), and hence write down the coordinates of \(A\).
- Find the coordinates of \(B\).
The shaded region \(R\) is bounded by the curve and the \(x\)-axis.
- Find the area of \(R\).