NYJC/2021/JC1/Promo/P1/Q05

(i) By writing $\frac{2-r}{r(r+1)(r+2)}$ in partial fractions, show that $\sum_{r=1}^n \frac{2-r}{r(r+1)(r+2)}=\frac{A n}{(n+1)(n+2)}$, where $A$ is a constant to be determined.

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(ii) Using the result in part (i), find $\sum_{r=1}^n \frac{1-r}{(r+1)(r+2)(r+3)}$ in terms of $n$.

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(iii) Hence find the exact value of $\sum_{r=10}^{\infty} \frac{1-r}{(r+1)(r+2)(r+3)}$.

[2]