Find $\sum\limits_{r=n}^{2n}{\left( {{3}^{r}}+n \right)}$.[3] Express \(\ln \left( \frac{{{r}^{2}}

Question

Find $\sum\limits_{r=n}^{2n}{\left( {{3}^{r}}+n \right)}$.[3] Express $\ln \left( \frac{{{r}^{2}}}{{{r}^{2}}-1} \right)$ as $A\ln \left( r-1 \right)+B\ln r+C\ln \left( r+1 \right)$, where $A$,$B$ and $C$ are integers to be determined. Hence, find $\sum\limits_{r=2}^{n}{\ln \left( \frac{{{r}^{2}}}{{{r}^{2}}-1} \right)}$, leaving your answer as a single logarithmic function in terms of $n$.[4]

Accepted Answer

(a) $\frac{3^n(3^{n+1} - 1)}{2} + n(n+1)$ (b) $\ln\frac{2n}{n+1}$

2019 EJC Promo Q5

2019 EJC Promo Q5

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