Express $\frac{4}{4{{r}^{2}}+16r+15}$ as $\frac{A}{2r+3}+\frac{B}{2r+5}$, where $A$

Question

Express $\frac{4}{4{{r}^{2}}+16r+15}$ as $\frac{A}{2r+3}+\frac{B}{2r+5}$, where $A$ and $B$ are constants to be determined. The sum $\sum\limits_{r=1}^{n}{\frac{4}{4{{r}^{2}}+16r+15}}$ is denoted by ${{S}_{n}}$.[1] Find an expression for ${{S}_{n}}$ in terms of $n$.[3] Find the smallest value of $n$ for which ${{S}_{n}}$ is within ${{10}^{-3}}$ of the sum to infinity.[3] Using he result in (ii), find $\sum\limits_{r=10}^{2n}{\frac{2}{4{{r}^{2}}-1}}$ in terms of $n$.[3]

Accepted Answer

(i) $A=2$, $B=-2$ (ii) $\dfrac{2}{5}-\dfrac{2}{2n+5}$ (iii) $998$ (iv) $\dfrac{1}{19}-\dfrac{1}{4n+1}$

2020 ASRJC P1 Q5

2020 ASRJC P1 Q5

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