Sequence and series

2020 RVHS P1 Q10

2020 RVHS P1 Q10 Sales agent $A$ started work on 1st June 2020. He plans to acquire $2$ clients in his first month of work and thereafter increase his clientele

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2018 SAJC P1 Q10

2018 SAJC P1 Q10 Albert and Betty each took a study loan of $$100,000$ from a bank on 1 January 2014 and both graduated on 31 December 2017. The bank

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2023 DHS Promo Q7

2023 DHS Promo Q7 (a) It is given that $sumlimits_{r=1}^{n}{{{r}^{2}}=frac{1}{6}nleft( n+1 right)left( 2n+1 right)}$. (i) Find $sumlimits_{r=1}^{n}{left( {{2}^{r+1}}+3r-{{r}^{2}} right)}$ in the form $Aleft( {{2}^{n}}-1 right)+text{f}left( n right)$, where $A$ is

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2011 NJC Q5 [Modified]

2011 NJC Q5 [Modified] A university student has a goal of saving at least $$1 000 000$ (in Singapore dollars). He begins working at the start of the year 2019.

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2023 NYJC MYE P1

2023 NYJC MYE P1 The series $a+ar-a{{r}^{2}}-a{{r}^{3}}+a{{r}^{4}}+a{{r}^{5}}-a{{r}^{6}}-a{{r}^{7}}+$…, where $a>0$, has its $k$th term, ${{T}_{k}}$, defined by ${{T}_{k}}=left{ begin{matrix}a{{r}^{k-1}}, \-a{{r}^{k-1}}, \end{matrix} right.$ $begin{matrix}text{if},k=4p-3,,text{or},,4p-2 \text{if},k=4p-1,,text{or},,4p,,,,,,, \end{matrix}$, for $pin {{mathbb{Z}}^{+}}$, (a) By rewriting

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2013 RI P1 Q9

2013 RI P1 Q9 Mr Tan decides to set up a scholarship fund for worthy students. On 1 January 2013, he places this scholarship fund in a bank investment which

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2023 HCI CT1 Q5 [Modified]

2023 HCI CT1 Q5 [Modified] (b) A sequence ${{a}_{1}}$, ${{a}_{2}}$, ${{a}_{3}}$, … is given by ${{a}_{1}}=frac{1}{2}$ and ${{a}_{n}}=frac{{{n}^{2}}}{{{2}^{n}}}$ for $nge 1$. Let ${{S}_{n}}$ be the sum of the first $n$

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2013 RI P1 Q8 [Modified]

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),

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NJC Sigma Notation Tutorial Q2

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

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NJC Sigma Notation Tutorial Q1

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

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