2022 CJC Promo Q4

It is known that the \({{n}^{\mathrm{th}}}\) term of a sequence is given by \({{u}_{n}}=p\left( {{4}^{-n}} \right)+qn+r\), where \(p\), \(q\) and \(r\) are constants.

It is given that \({{u}_{1}}=6\), \({{u}_{2}}=0\) and \({{u}_{3}}=-\frac{15}{4}\).

1. Find \(p\), \(q\) and \(r\).[3]
2. Show \(\sum\limits_{r=1}^{n}{{{u}_{r}}}=A+B\left( {{4}^{-n}} \right)+Cn+D{{n}^{2}}\) where \(A\), \(B\), \(C\) and \(D\) are constants to be determined.[4]