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2022 DHS Promo Q10 (a) By considering $4r-1=5r-left( 1+r right)$, use the method of differences to find $sumlimits_{r=1}^{n}{left( frac{4r-1}{{{5}^{r}}} right)}$. [4] (b) Hence find $sumlimits_{r=1}^{infty }{left( frac{4r-1}{{{5}^{r}}} right)}$, justifying your
2022 ASRJC P1 Q5 (i) By considering ${{u}_{n}}-{{u}_{n+1}}$, where ${{u}_{n}}=frac{1}{nleft( n+1 right)left( n+2 right)}$, find $sumlimits_{n=1}^{N}{frac{1}{nleft( n+1 right)left( n+2 right)left( n+3 right)}}$ in terms of $N$. [3] (ii) Hence or
2022 CJC Promo Q4 It is known that the ${{n}^{text{th}}}$ term of a sequence is given by ${{u}_{n}}=pleft( {{4}^{-n}} right)+qn+r$, where $p$, $q$ and $r$ are constants. It is given
2020 EJC P1 Q7 The sum, ${{S}_{n}}$, of the first $n$ terms of a sequence ${{u}_{1}}$, ${{u}_{2}}$, ${{u}_{3}}$, … is given by ${{S}_{n}}=aleft( {{3}^{n}} right)+bn+c$, where $a$, $b$ and $c$
2022 RI P1 Q6 (a) A sequence is such that ${{u}_{1}}=p$, where $p$ is a constant, and ${{u}_{n+1}}=frac{4}{{{u}_{n}}}$, for $n>0$. Describe how the sequence behaves when (i) $p=2$, [1] (ii)
Home 2023 CJC Sigma Notation Tutorial Q7 Find the values of $A$, $B$ and $C$ such that for all values of $r$, $1+{{r}^{2}}=Aleft( r+2 right)left( r+1 right)+Bleft( r+1 right)+C$. Hence,
Home 2013 JJC P1 Q3 (i) Show that $frac{1}{sqrt{r+1}+sqrt{r}}=sqrt{r+1}-sqrt{r}$ for all $rin {{mathbb{Z}}^{+}}$. (ii) Hence find $sumlimits_{r=1}^{n}{frac{1}{sqrt{r+1}+sqrt{r}}}$. (iii) Deduce that $sumlimits_{r=1}^{n}{frac{1}{sqrt{r}}>2left( sqrt{n+1}-1 right)}$. Suggested Video Solutions Suggested Handwritten Solutions (i)
Home 2018 NYJC Promo Q6 (i) Show that $frac{1}{left( r-1 right)rleft( r+1 right)}$ can be expressed as $frac{A}{r-1}+frac{B}{r}+frac{C}{r+1}$, where $A$, $B$ and $C$ are constants to be determined. [1] (ii)
JJC P1 Q13 (i) Express $frac{1}{left( 2x+1 right)left( 2x+3 right)}$ in partial fractions. (ii) Hence, show that $sumlimits_{r=1}^{n}{frac{1}{left( 2r+1 right)left( 2r+3 right)}}=frac{n}{3left( 2n+3 right)}$. (iii) Deduce the sum of $frac{1}{3}+frac{1}{3left(
Home SAJC Sigma Notation Tutorial Q1 By considering $text{f}left( r right)-text{f}left( r+1 right)$ where $text{f}left( r right)=ln left( 1+frac{1}{r} right)$, find ${{S}_{n}}$, the sum of the first $n$ terms of