N2024 P1 Q5

A geometric series has first term \(a\) and common ratio \(r\), where \(r<0\). The sum to infinity of the series is \(18\) and its third term is \(\frac{8}{3}\).

1. Show that \(27{{r}^{3}}-27{{r}^{2}}+4=0\).[3]
2. Find the values of \(r\) and \(a\).[1]

The \(n\) th term of the series is \({{u}_{n}}\).

3. It is given that
   \(80\sum\limits_{r=k+1}^{\infty }{{{u}_{r}}}=\sum\limits_{r=1}^{k}{{{u}_{r}}}\).
   Find the value of \(k\).[3]