2017 TJC P1 Q5

The curve \(C\) has equation\(y=k{{x}^{3}}\) . The tangent at the point \(P\) on \(C\) meets the curve again at point \(Q\). The tangent at point \(Q\) meets the curve again at point \(R\). If the \(x\) coordinates of \(P\), \(Q\) and \(R\) are \(p\) , \(q\), and \(r\) respectively where \(p\ne 0\) .

1. Show that \(p\) and \(q\) satisfy the equation \({{\left( \frac{q}{p} \right)}^{2}}+\left( \frac{q}{p} \right)-2=0\) .[4]
2. Show that \(p\) , \(q\), and \(r\) are three consecutive terms of a geometric progression. Hence determine if this geometric series is convergent.[4]

[You may use the identity \({{a}^{3}}-{{b}^{3}}=\left( a-b \right)\left( {{a}^{2}}+ab+{{b}^{2}} \right)\) for \(a,b\in \mathbb{R}\).]