Sigma Notation

Home 2018 DHS Promo Q3  It is given that $text{f}left( r right)=2{{r}^{3}}-3{{r}^{2}}+r-4$. (i) Show that $text{f}left( r+1 right)-text{f}left( r right)=k{{r}^{2}}$, where $k$ is a constant to be determined. [2] (ii)

Home 2022 HCI J1 BT Q7 It is given that ${{u}_{r}}=frac{1}{left( r+2k right)!}$, where $rin {{mathbb{Z}}^{+}}$ and $k$ is a positive constant. (i) Show that ${{u}_{r}}-{{u}_{r+1}}=frac{r+2k}{left( r+2k+1 right)!}$. [1] (ii)

Home 2022 VJC J1 MYE Q5 (a) A sequence ${{u}_{1}}$, ${{u}_{2}}$, ${{u}_{3}}$, … is such that ${{u}_{n+2}}=A{{u}_{n+1}}-{{u}_{n}}$, where $A$ is a positive constant and $nge 1$. Given that ${{u}_{1}}=1$, ${{u}_{3}}=5$

2022 CJC J1 MYE Q6 (i) Find $sumlimits_{r=1}^{n}{left( {{3}^{r}}+16r-8 right)}$, giving your answer in the form $Aleft( {{3}^{n}}-1 right)+B{{n}^{2}}$, where $A$ and $B$ are constants to be determined. [4] (ii)

Home 2016 HCI P1 Q4 Prove that $frac{2n+1}{sqrt{{{n}^{2}}+2n}+sqrt{{{n}^{2}}-1}}=sqrt{{{n}^{2}}+2n}-sqrt{{{n}^{2}}-1}$. [2] Hence find $sumlimits_{n=1}^{N}{frac{2n+1}{sqrt{{{n}^{2}}+2n}+sqrt{{{n}^{2}}-1}}}$. [3] (a) Deduce the value of $sumlimits_{n=2}^{N}{frac{2n-1}{sqrt{{{n}^{2}}-2n}+sqrt{{{n}^{2}}-1}}}$. [3] (b) Show that $sumlimits_{n=1}^{N}{frac{2n+1}{2n-1}}>sqrt{{{N}^{2}}+2N}$. [1] Suggested Video and Handwritten Solutions

2022 ACJC P2 Q4 (i) It is given that ${{U}_{n}}=cos left[ left( 2n+1 right)theta right]$, for $nge 0$.Show that for $nge 1$, ${{U}_{n}}+{{U}_{n-1}}=2cos left( 2ntheta right)cos theta $. [1] (ii)

2022 HCI P2 Q4 (a) (i) Find $sumlimits_{r=1}^{k}{left[ {{left( -frac{1}{2} right)}^{r+1}}+ln left( r+1 right) right]}$ in terms of $k$. Simplify your answer. [4] (ii) Hence determine if $sumlimits_{r=1}^{infty }{left[ {{left(

2022 HCI P1 Q3 It is given that ${{u}_{r}}=frac{1}{r!}$, $rin {{mathbb{Z}}^{+}}$. (i) Show that ${{u}_{r}}-2{{u}_{r+1}}+{{u}_{r+2}}=frac{{{r}^{2}}+r-1}{left( r+2 right)!}$. [1] (ii) Hence find $sumlimits_{r=1}^{n}{frac{{{r}^{2}}+r-1}{left( r+2 right)!}}$ in terms of $n$ and determine