2025 ASRJC P1 Q6

It is given that\(\sum\limits_{r=1}^{n}{\frac{7r+4}{r\left( r+1 \right)\left( r+2 \right)}}=\frac{9}{2}-\frac{2}{n+1}-\frac{5}{n+2}\)

1. Find \(\sum\limits_{r=7}^{2n}{\frac{7r-3}{{{r}^{3}}-r}}\), giving your answers in terms of \(n\).[4]
2. Show algebraically that \({{\left( r+1 \right)}^{3}}>r\left( r+1 \right)\left( r+2 \right)\) for all positive integers \(r\).[2]
3. Hence show that \(\sum\limits_{r=1}^{n}{\frac{7r+4}{{{\left( r+1 \right)}^{3}}}}<\frac{9}{2}\).[3]