2018 EJC Promo Q8
(i)
Show that $\frac{4}{\left( r-1 \right)!}-\frac{1}{r!}-\frac{3}{\left( r+1 \right)!}=\frac{4{{r}^{2}}+3r-4}{\left( r+1 \right)!}$ .
Hence, express $\sum\limits_{r=1}^{n}{\frac{4{{r}^{2}}+3r-4}{\left( r+1 \right)!}}$ in the form $A-\text{f}\left( n \right)$, where $A$ is a constant and $\text{f}\left( n \right)$ is a single fraction in terms of $n$.
[5]
(ii)
Deduce that $\sum\limits_{r=1}^{n}{\frac{{{r}^{2}}-1}{\left( r+1 \right)!}<\frac{7}{4}}$ .
[3]
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