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PSLE Foundation Mathematics

<ol class="abc-list"><li data-list-item-id="asrjc-q10-item01" data-list-item-path="asrjc-q10-item01"><div class="list-item-paragraph">Given that&nbsp;$k$ is a real constant, find $\displaystyle \sum_{r=5}^{30}[2^{1-2r}-(k-2r)]$, leaving your answer in the form $A(1-B)+Ck+910$, where&nbsp;$A$, $B$ and $C$ are exact real constants.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="asrjc-q10-item02" data-list-item-path="asrjc-q10-item02"><div class="list-item-paragraph">Another series is such that \[\sum_{r=1}^{N}\frac1{r(r+1)(r+2)}=\frac12\left(\frac12-\frac1{(N+1)(N+2)}\right).\]</div><ol class="i-list"><li data-list-item-id="asrjc-q10-item02-sub-item01" data-list-item-path="asrjc-q10-item02/asrjc-q10-item02-sub-item01"><div class="list-item-paragraph">Find $\displaystyle \sum_{r=1}^{\infty}\frac1{r(r+1)(r+2)}$. Explain your answer clearly.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="asrjc-q10-item02-sub-item02" data-list-item-path="asrjc-q10-item02/asrjc-q10-item02-sub-item02"><div class="list-item-paragraph">Find $\displaystyle \sum_{r=13}^{N}\frac1{3r(r-1)(r-2)}$ in terms of $N$.<span class="inline-marks">[3]</span></div></li></ol></li><li data-list-item-id="asrjc-q10-item03" data-list-item-path="asrjc-q10-item03"><div class="list-item-paragraph">A sequence of positive numbers $x_1,x_2,x_3,\ldots$ satisfies the recurrence relation \[x_{n+1}=\sqrt{10-3x_n},\qquad n=1,2,3,\ldots\]</div><ol class="i-list"><li data-list-item-id="asrjc-q10-item03-sub-item01" data-list-item-path="asrjc-q10-item03/asrjc-q10-item03-sub-item01"><div class="list-item-paragraph">Given that&nbsp;$x_4=1$, find the exact value of $x_1$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="asrjc-q10-item03-sub-item02" data-list-item-path="asrjc-q10-item03/asrjc-q10-item03-sub-item02"><div class="list-item-paragraph">It is also known that the sequence converges to $l$. Find the value of $l$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="asrjc-q10-item03-sub-item03" data-list-item-path="asrjc-q10-item03/asrjc-q10-item03-sub-item03"><div class="list-item-paragraph">Prove that&nbsp;$(x_{n+1})^2-l^2=3(l-x_n)$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="asrjc-q10-item03-sub-item04" data-list-item-path="asrjc-q10-item03/asrjc-q10-item03-sub-item04"><div class="list-item-paragraph"><p>Use the result in part <strong>(iii)</strong> to show that if $x_n&lt;l$, then $x_{n+1}&gt;l $.</p><span class="inline-marks">[2]</span></div></li></ol></li></ol>

Given that $k$ is a real constant, find $\displaystyle \sum_{r=5}^{30}[2^{1-2r}-(k-2r)]$, leaving your answer in the form $A(1-B)+Ck+910$, where $A$, $B$ and $C$ are exact real constants.

Sigma Notation By Substitution

Limit of a Sequence and Behaviour of a Sequence

Manipulating Summation Limits

Application of Sum to Infinity

Reverse Iteration of Recurrence Relations

Summations Involving APGP

2026 ASRJC P1 Q10

Sequence and Series

<p>A sequence of real numbers $x_1,x_2,x_3,\ldots$ satisfies the recurrence relation</p>
<p style="text-align: center">\[x_{n+1}=\frac{4x_n+3}{x_n+2}\quad\text{for all}\hspace{0.5em}n\geq1.\]</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Given that the sequence converges to $l$, find the possible exact values of $l$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">Describe how the sequence behaves when&nbsp;$x_1=2$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">Given that&nbsp;$x_4=\frac{187}{63}$, find the value of $x_1$.<span class="inline-marks">[3]</span></div></li></ol>

A sequence of real numbers $x_1,x_2,x_3,\ldots$ satisfies the recurrence relation

\[x_{n+1}=\frac{4x_n+3}{x_n+2}\quad\text{for all }n\geq1.\]

Convergence and Reverse Iteration of a Recurrence

<p>A sequence is such that&nbsp;$u_1 = N$ where&nbsp;$N$ is a constant, and $u_{n+1} = 3u_n - 8$, for&nbsp;$n \ge 1$.</p>
<ol class="abc-list"><li data-list-item-id="item-seq-a" data-list-item-path="item-seq-a"><div class="list-item-paragraph">Describe how the sequence behaves when</div><ol class="i-list"><li data-list-item-id="item-seq-a-i" data-list-item-path="item-seq-a/item-seq-a-i"><div class="list-item-paragraph">$N = 4$,<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-seq-a-ii" data-list-item-path="item-seq-a/item-seq-a-ii"><div class="list-item-paragraph">$N = 3$.<span class="inline-marks">[1]</span></div></li></ol></li><li data-list-item-id="item-seq-b" data-list-item-path="item-seq-b"><div class="list-item-paragraph">Find the value of $N$ for which $u_5 = 409$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-seq-c" data-list-item-path="item-seq-c"><div class="list-item-paragraph">The sequence $v_n = (-1)^n (u_n + k)$ for&nbsp;$n \ge 1$ where&nbsp;$k$ is a constant, is convergent when&nbsp;$u_1 = 4$. State the value of $k$.<span class="inline-marks">[1]</span></div></li></ol>

A sequence is such that $u_1 = N$ where $N$ is a constant, and $u_{n+1} = 3u_n - 8$, for $n \ge 1$.

Find the value of $N$ for which $u_5 = 409$.

The sequence $v_n = (-1)^n (u_n + k)$ for $n \ge 1$ where $k$ is a constant, is convergent when $u_1 = 4$. State the value of $k$.

Inter-sequence Relationships

2025 RVHS P2 Q1

<p>A sequence of real numbers $x_1$, $x_2$, $x_3$, $\dots$ satisfies the recurrence relation</p>
<p style="text-align: center">\[x_{n+1} = \frac{5x_n + 2}{2x_n + 3} \text{ for all}\hspace{0.5em} n \ge 1.\]</p>
<ol class="abc-list"><li data-list-item-id="item-a-3"><div class="list-item-content"><span class="list-item-text">Given that the sequence converges to $l$, find the possible exact values of $l$.</span><span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-b-3"><div class="list-item-content"><span class="list-item-text">Describe how the sequence behaves when&nbsp;$x_1 = 3$.</span><span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-c-3"><div class="list-item-content"><span class="list-item-text">Given that&nbsp;$x_5 = \frac{3503}{2158}$, find the value of $x_1$.</span><span class="inline-marks">[3]</span></div></li></ol>

A sequence of real numbers $x_1$, $x_2$, $x_3$, $\dots$ satisfies the recurrence relation

\[x_{n+1} = \frac{5x_n + 2}{2x_n + 3} \text{ for all } n \ge 1.\]

Given that the sequence converges to $l$, find the possible exact values of $l$.

Describe how the sequence behaves when $x_1 = 3$.

Given that $x_5 = \frac{3503}{2158}$, find the value of $x_1$.

2025 DHS P1 Q4 

<p>A sequence ${{t}_{1}},{{t}_{2}},{{t}_{3}},{{t}_{4}},\ldots $ is given by</p>
<p style="text-align: center">${{t}_{n+1}}=a+b{{t}_{n}}$,</p>
<p>where&nbsp;$a$ and $b$ are non-zero constants.</p>
<p>It is given that&nbsp;${{t}_{3}}=320$, ${{t}_{4}}=240$ and ${{t}_{5}}=200$.</p>
<ol class="abc-list"><li data-list-item-id="item-1760870146720-s1k6gagvx" data-list-item-path="item-1760870146720-s1k6gagvx"><div class="list-item-paragraph">Determine the value of $a$ and the value of $b$.</div></li><li data-list-item-id="item-1760870147342-aqwsh8ard" data-list-item-path="item-1760870147342-aqwsh8ard"><div class="list-item-paragraph">Find the value of ${{t}_{6}}$.</div></li><li data-list-item-id="item-1760870148010-7e2lh3d3k" data-list-item-path="item-1760870148010-7e2lh3d3k"><div class="list-item-paragraph">Show clearly that&nbsp;${{t}_{1}}=800$.</div></li></ol>
<p>The sequence converges to a limit $L$.</p>
<ol class="abc-list" start="4"><li data-list-item-id="item-1760870172066-7t3qt9pe6" data-list-item-path="item-1760870172066-7t3qt9pe6"><div class="list-item-paragraph">Determine the value of $L$.</div></li></ol>

A sequence ${{t}_{1}},{{t}_{2}},{{t}_{3}},{{t}_{4}},\ldots $ is given by

where $a$ and $b$ are non-zero constants.

It is given that ${{t}_{3}}=320$, ${{t}_{4}}=240$ and ${{t}_{5}}=200$.

Generating Terms from Recurrence Relations

Determining Model Coefficients Using Linear Systems

T.Madas - Recurrence Relations Q37

System of Linear Equations

<p>During a clinical trial, the concentration ${{U}_{t}}$ of a specific blood agent is measured at one-hour intervals following the initial administration of the trial drug to a patient.</p>
<p>The following readings were obtained:</p>
<p style="text-align: center">${{U}_{3}}=88$, ${{U}_{4}}=76$ and ${{U}_{5}}=70$,</p>
<p>where&nbsp;${{U}_{t}}$ denotes the reading $t$ hours after the drug was first administered.</p>
<p>It is believed that&nbsp;$U$ satisfies the relationship</p>
<p style="text-align: center">${{U}_{t+1}}=p+q{{U}_{t}}$, $t\ge 0$,</p>
<p>where&nbsp;$p$ and $q$ are constants.</p>
<ol class="abc-list"><li data-list-item-id="item-1760867603859-k9t1qqbhq" data-list-item-path="item-1760867603859-k9t1qqbhq"><div class="list-item-paragraph">Find the values of $p$ and $q$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1760867604504-menwt8l4h" data-list-item-path="item-1760867604504-menwt8l4h"><div class="list-item-paragraph">Determine the initial concentration of the blood agent when the drug was initially administered.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1760867605591-rovuqf55s" data-list-item-path="item-1760867605591-rovuqf55s"><div class="list-item-paragraph">Describe how the concentration behaves over a long period of time.<span class="inline-marks">[1]</span></div></li></ol>

During a clinical trial, the concentration ${{U}_{t}}$ of a specific blood agent is measured at one-hour intervals following the initial administration of the trial drug to a patient.

${{U}_{3}}=88$, ${{U}_{4}}=76$ and ${{U}_{5}}=70$,

where ${{U}_{t}}$ denotes the reading $t$ hours after the drug was first administered.

It is believed that $U$ satisfies the relationship

2024 NYJC Promo Q4

<ol class="abc-list"><li><div class="list-item-paragraph">A sequence is such that ${{u}_{1}}=p$, where $p$ is a constant and ${{u}_{n+1}}=\frac{5{{u}_{n}}}{8{{u}_{n}}+1}$, for $n\ge 1$.</div><ol class="i-list"><li><div class="list-item-paragraph">Describe how the sequence behaves when $p=1$.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Find the value of $p$ for which ${{u}_{6}}=\frac{3125}{6253}$.</div><p style="text-align: right;">[2]</p></li></ol></li><li><div class="list-item-paragraph">Another sequence ${{v}_{1}}$, ${{v}_{2}}$, ${{v}_{3}}$, … is such that for all $n\ge 1$, ${{v}_{n+2}}-2{{v}_{n+1}}+{{v}_{n}}=k$, where $k$ is a constant.</div><div class="list-item-paragraph">Let ${{w}_{n}}={{v}_{n+1}}-{{v}_{n}}$ for $n\ge 1$. Explain why the sequence $\left\{ {{w}_{n}} \right\}$ is an arithmetic progression.</div><p style="text-align: right;">[2]</p></li></ol>

Proving the progression

2025 VJC P1 Q4

<p>A sequence is such that ${{u}_{1}}=p$, where $p$ is a constant, and ${{u}_{n+1}}=2{{u}_{n}}-5$, for $n > 0$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Describe how the sequence behaves when</div><ol class="abc-list"><li><div class="list-item-paragraph">$p=7$,</div><p style="text-align: right;">[1]</p></li><li><div class="list-item-paragraph">$p=5$.</div><p style="text-align: right;">[1]</p></li></ol></li><li><div class="list-item-paragraph">Find the value of $p$ for which ${{u}_{5}}=101$.</div><p style="text-align: right;">[2]</p></li></ol>

A sequence is such that ${{u}_{1}}=p$, where $p$ is a constant, and ${{u}_{n+1}}=2{{u}_{n}}-5$, for $n > 0$.

RI Sequences and Series Tutorial 7A Q2

<p>A sequence ${{u}_{1}}$, ${{u}_{2}}$, ${{u}_{3}}$, … is such that&nbsp;${{u}_{n+2}}+a{{u}_{n+1}}+{{u}_{n}}=b$, where&nbsp;$a$, $b$ are constants and $n\ge 1$.</p>
<ol class="i-list"><li data-list-item-id="item-1759645347598-0wjcvoqiy" data-list-item-path="item-1759645347598-0wjcvoqiy"><div class="list-item-paragraph">Given that&nbsp;${{u}_{1}}=c$, ${{u}_{2}}=0.5$, ${{u}_{3}}=5$, ${{u}_{4}}=14.5$ and ${{u}_{5}}=29$, where&nbsp;$c$ is a constant, find $a$, $b$ and $c$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1759645349001-144kb5st6" data-list-item-path="item-1759645349001-144kb5st6"><div class="list-item-paragraph">By using the substitution ${{v}_{n}}={{u}_{n+1}}-{{u}_{n}}$, explain why the sequence $\left\{ {{v}_{n}} \right\}$ is an arithmetic progression.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1759645349707-bohkaxg4g" data-list-item-path="item-1759645349707-bohkaxg4g"><div class="list-item-paragraph">By considering $\sum\limits_{r=1}^{n-1}{{{v}_{r}}=\sum\limits_{r=1}^{n-1}{\left( {{u}_{r+1}}-{{u}_{r}} \right)}}$, find ${{u}_{n}}$ in terms of $n$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-1759645350389-3qzarkse7" data-list-item-path="item-1759645350389-3qzarkse7"><div class="list-item-paragraph">Describe the behaviour of the sequence $\left\{ {{u}_{n}} \right\}$ for large values of $n$.<span class="inline-marks">[1]</span></div></li></ol>

A sequence ${{u}_{1}}$, ${{u}_{2}}$, ${{u}_{3}}$, … is such that ${{u}_{n+2}}+a{{u}_{n+1}}+{{u}_{n}}=b$, where $a$, $b$ are constants and $n\ge 1$.

Method of Difference

Arithmetic Progression

2021 DHS P1 Q7

<ol class="abc-list"><li data-list-item-id="item-1759490768900-oqdcr3xj9"><div class="list-item-paragraph">A sequence ${{u}_{1}}$, ${{u}_{2}}$, ${{u}_{3}}$, … is such that&nbsp;${{u}_{n+2}}=A{{u}_{n+1}}-{{u}_{n}}$, where&nbsp;$A$ is a positive constant and $n\ge 1$. Given that&nbsp;${{u}_{1}}=1$, ${{u}_{3}}=5$ and ${{u}_{4}}=13$, find $A$ and ${{u}_{2}}$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-1759490769695-vs1gpadz2"><div class="list-item-paragraph">The sum of the first $n$ terms of a sequence ${{v}_{1}}$, ${{v}_{2}}$, ${{v}_{3}}$, … is given by $2{{n}^{2}}+4n$.</div><ol class="i-list"><li data-list-item-id="subitem-1759490785895-s48rqdy57"><div class="list-item-paragraph">Find an expression for&nbsp;${{v}_{n}}$, in terms of $n$, in simplified form.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="subitem-1759490786083-jtivpngul"><div class="list-item-paragraph">Hence find ${{v}_{2}}+{{v}_{4}}+{{v}_{6}}+...+{{v}_{48}}$.<span class="inline-marks">[3]</span></div></li></ol></li></ol>

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 $n\ge 1$. Given that ${{u}_{1}}=1$, ${{u}_{3}}=5$ and ${{u}_{4}}=13$, find $A$ and ${{u}_{2}}$.

The sum of the first $n$ terms of a sequence ${{v}_{1}}$, ${{v}_{2}}$, ${{v}_{3}}$, … is given by $2{{n}^{2}}+4n$.

Subsequences of Progressions

$u_n = S_n - S_{n-1} $

2022 VJC J1 MYE Q5

<ol class="abc-list"><li data-list-item-id="item-1758966125414-yu4ro77z7" data-list-item-path="item-1758966125414-yu4ro77z7"><div class="list-item-paragraph">A sequence is such that&nbsp;${{u}_{1}}=p$ where&nbsp;$p$ is a constant and ${{u}_{n+1}}=2{{u}_{n}}-5$, for&nbsp;$n&gt;0$.</div><ol class="i-list"><li data-list-item-id="subitem-1758966133140-e2hnl0c3v" data-list-item-path="item-1758966125414-yu4ro77z7/subitem-1758966133140-e2hnl0c3v"><div class="list-item-paragraph">Describe how the sequence behaves when</div><div class="list-item-paragraph">(A) $p=7$,<span class="inline-marks">[1]</span></div><div class="list-item-paragraph">(B) $p=5$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="subitem-1758966182339-yto1stauw" data-list-item-path="item-1758966125414-yu4ro77z7/subitem-1758966182339-yto1stauw"><div class="list-item-paragraph">Find the value of $p$ for which ${{u}_{5}}=101$.<span class="inline-marks">[2]</span></div></li></ol></li><li data-list-item-id="item-1758966216464-vh6jv3st2" data-list-item-path="item-1758966216464-vh6jv3st2"><div class="list-item-paragraph">Another sequence is defined by ${{v}_{1}}=a$, ${{v}_{2}}=b$, where&nbsp;$a$ and $b$ are constants, and ${{v}_{n+2}}={{v}_{n}}+2{{v}_{n+1}}-7$, for&nbsp;$n&gt;0$.</div><div class="list-item-paragraph">For this sequence, ${{v}_{4}}=2{{v}_{3}}$.</div><ol class="i-list"><li data-list-item-id="subitem-1758966220327-la1gw58yh" data-list-item-path="item-1758966216464-vh6jv3st2/subitem-1758966220327-la1gw58yh"><div class="list-item-paragraph">Find the value of $b$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="subitem-1758966220739-sz4cv4pzn" data-list-item-path="item-1758966216464-vh6jv3st2/subitem-1758966220739-sz4cv4pzn"><div class="list-item-paragraph">Find an expression in terms of $a$ for&nbsp;${{v}_{5}}$.<span class="inline-marks">[1]</span></div></li></ol></li><li data-list-item-id="item-1758967997887-5h072gg0d" data-list-item-path="item-1758967997887-5h072gg0d"><div class="list-item-paragraph">The sum of the first $n$ terms of a series is ${{n}^{3}}-11{{n}^{2}}+4n$, where&nbsp;$n$ is a positive integer.</div><ol class="i-list"><li data-list-item-id="subitem-1758967999327-3acnlia3x" data-list-item-path="item-1758967997887-5h072gg0d/subitem-1758967999327-3acnlia3x"><div class="list-item-paragraph">Find an expression for the $n$th term of this series, giving your answer in its simplest form.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="subitem-1758967999492-e9b8e0b0j" data-list-item-path="item-1758967997887-5h072gg0d/subitem-1758967999492-e9b8e0b0j"><div class="list-item-paragraph">The sum of the first $m$ terms of this series, where&nbsp;$m&gt;3$, is equal to the sum of the first three terms of this series. Find the value of $m$.<span class="inline-marks">[2]</span></div></li></ol></li></ol>

A sequence is such that ${{u}_{1}}=p$ where $p$ is a constant and ${{u}_{n+1}}=2{{u}_{n}}-5$, for $n>0$.

Another sequence is defined by ${{v}_{1}}=a$, ${{v}_{2}}=b$, where $a$ and $b$ are constants, and ${{v}_{n+2}}={{v}_{n}}+2{{v}_{n+1}}-7$, for $n>0$.

The sum of the first $n$ terms of a series is ${{n}^{3}}-11{{n}^{2}}+4n$, where $n$ is a positive integer.

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