CJC/2021/JC1/Promo/P1/Q05

(a) (i) Express $\frac{2 r^2+1}{r^2-1}$ in the form $A+\frac{B}{r-1}+\frac{C}{r+1}$, where $A, B$ and $C$ are constants to be found.

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(ii) Hence find $\sum_{r=2}^n \frac{2 r^2+1}{r^2-1}$ in terms of $n$.

[4]

(b) Express $\sum_{r=1}^{2 n}\left[(-1)^{r+1} 2^r\right]$ in the form $\frac{p}{q}\left[s\left(4^n\right)+t\right]$, where $p, q$, $s$ and $t$ are integers to be determined.

[3]