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2010 NYJC Promo Q5 [Modified] (i) Using partial fractions, show that $sumlimits_{n=1}^{N}{frac{1}{nleft( n+1 right)left( n+2 right)}=frac{1}{4}-frac{1}{2left( N+1 right)left( N+2 right)}}$. [4] (ii) Deduce the value of $sumlimits_{n=2}^{infty }{frac{1}{nleft( n+1 right)left(
2022 YIJC Promo Q4 (a) Express $frac{5r+8}{rleft( r+1 right)left( r+2 right)}$ in partial fractions. [2] (b) Hence, find $sumlimits_{r=,,1}^{n}{frac{5r+8}{rleft( r+1 right)left( r+2 right)}}$, giving your answer in the form $M-frac{P}{n+1}-frac{Q}{n+2}$,
2022 RI Promo Q4 (i) Find $sumlimits_{r=1}^{n}{frac{1}{left( r+1 right)left( r+3 right)}}$, where $nge 3$. (There is no need to express your answer as a single algebraic fraction.) [5] (ii) Explain
2019 SAJC P2 Q4 (a) Given that $text{f}left( r right)=frac{r}{2{{,}^{r}}}$, by considering $text{f}left( r+1 right)-text{f}left( r right)$, find $sumlimits_{r=,1}^{n}{frac{1-r}{2{{,}^{r+1}}}}$. [3] (b) (i) Cauchy’s root test states that a series of
2023 CJC J2 MYE P1 Q3 A sequence ${{u}_{0}}$, ${{u}_{1}}$, ${{u}_{2}}$, … is defined by ${{u}_{0}}=10$ and ${{u}_{n}}={{u}_{n-1}}+A{{n}^{2}}+Bn$ where $A$ and $B$ are constants and $nge 1$. It is further
2015 VJC P2 Q7 A game is to be played between two players, $A$ and $B$. A bag contains $5$ balls each with $A$’s name and $8$ balls each with
2009 SAJC P2 Q3 [Modified] Given that ${{u}_{r}}=frac{1}{{{r}^{3}}}$, (i) Show that ${{u}_{r-1}}-{{u}_{r+1}}=frac{6{{r}^{2}}+2}{{{left( {{r}^{2}}-1 right)}^{3}}}$. (ii) Hence find $sumlimits_{r=2}^{n}{frac{3{{r}^{2}}+1}{{{left( {{r}^{2}}-1 right)}^{3}}}}$ in terms of $n$. (iii) Use your answer to part
2013 RI P1 Q8 [Modified] (i) Show that ${{sin }^{4}}A={{sin }^{2}}A-frac{1}{4}{{sin }^{2}}2A$. It is given that ${{S}_{n}}=sumlimits_{r=0}^{n}{frac{1}{{{4}^{r}}}{{sin }^{4}}left[ {{2}^{r}}left( frac{pi }{3} right) right]}$. (ii) By using the result in (i),
NJC Sigma Notation Tutorial Q2 (i) Prove that for all positive integers $n$, $frac{1}{n!}-frac{1}{left( n+1 right)!}=frac{1}{n!+left( n-1 right)!}$. (ii) Hence evaluate $sumlimits_{n=1}^{N}{frac{1}{n!+left( n-1 right)!}}$ in terms of $N$. (iii) By
NJC Sigma Notation Tutorial Q1 The method of differences can be used to evaluate $sumlimits_{r=1}^{n}{text{f}left( r right)}$, where $text{f}left( r right)=frac{2}{r}-frac{3}{r+1}+frac{1}{r+2}$. In each of the following situations, explain clearly the