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<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.

Limit of a Sequence and Behaviour of a Sequence

Method of Difference

Sigma Notation

2026 ASRJC P1 Q10

Sequence and Series

<p>Mr Tan decides to start a regular savings plan to save up for an overseas trip. At the start of the first month, he deposits $1000 and continues to deposit $800 at the start of every subsequent month. He withdraws $10\%$ of the amount in the account at the end of each month for investment starting from the first month. The account pays no interest.</p>
<ol class="abc-list"><li data-list-item-id="h2-2025-nyjc-promo-q8-nyjc-q8-item-block-1-item-0" data-list-item-path="h2-2025-nyjc-promo-q8-nyjc-q8-item-block-1-item-0"><div class="list-item-paragraph">Show that the amount of money in the savings account at the end of the $n$th month is $\$(7200 - 7000(0.9)^n)$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="h2-2025-nyjc-promo-q8-nyjc-q8-item-block-1-item-1" data-list-item-path="h2-2025-nyjc-promo-q8-nyjc-q8-item-block-1-item-1"><div class="list-item-paragraph">Find the earliest possible month such that the amount of money in the savings account first reaches $5000. Explain whether this occurs at the start or end of the month.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="h2-2025-nyjc-promo-q8-nyjc-q8-item-block-1-item-2" data-list-item-path="h2-2025-nyjc-promo-q8-nyjc-q8-item-block-1-item-2"><div class="list-item-paragraph">State the amount of money that Mr Tan has in the savings account in the long run after withdrawing for his monthly investment.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="h2-2025-nyjc-promo-q8-nyjc-q8-item-block-1-item-3" data-list-item-path="h2-2025-nyjc-promo-q8-nyjc-q8-item-block-1-item-3"><div class="list-item-paragraph">Find the total amount of money Mr Tan withdraws for investment at the end of 1 year.<span class="inline-marks">[2]</span></div></li></ol>

Mr Tan decides to start a regular savings plan to save up for an overseas trip. At the start of the first month, he deposits $1000 and continues to deposit $800 at the start of every subsequent month. He withdraws $10\%$ of the amount in the account at the end of each month for investment starting from the first month. The account pays no interest.

Show that the amount of money in the savings account at the end of the $n$th month is $\$(7200 - 7000(0.9)^n)$.

Find the earliest possible month such that the amount of money in the savings account first reaches $5000. Explain whether this occurs at the start or end of the month.

State the amount of money that Mr Tan has in the savings account in the long run after withdrawing for his monthly investment.

Find the total amount of money Mr Tan withdraws for investment at the end of 1 year.

Problems involving inequalities

Geometric Progression

Financial Arithmetic or Iterative Pattern Recognition

2025 NYJC Promo Q8

<p>The sequence $x_1, x_2, x_3, \ldots$ satisfies the recurrence relation \[x_{n+1}=x_n^2(2-x_n)\text{ for}\hspace{0.5em}n\geq 1\]</p>
<ol class="abc-list"><li data-list-item-id="h2-2025-ejc-promo-q5-ejc-q5-item-block-2-item-0" data-list-item-path="h2-2025-ejc-promo-q5-ejc-q5-item-block-2-item-0"><div class="list-item-paragraph">Prove algebraically that, if the sequence converges, then it converges to either 1 or 0.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="h2-2025-ejc-promo-q5-ejc-q5-item-block-2-item-1" data-list-item-path="h2-2025-ejc-promo-q5-ejc-q5-item-block-2-item-1"><div class="list-item-paragraph">Determine the behaviour of the sequence for each of the following cases.</div><ol class="i-list"><li data-list-item-id="h2-2025-ejc-promo-q5-ejc-q5-item-block-2-item-1-sub-0" data-list-item-path="h2-2025-ejc-promo-q5-ejc-q5-item-block-2-item-1/h2-2025-ejc-promo-q5-ejc-q5-item-block-2-item-1-sub-0"><div class="list-item-paragraph">$x_1=0.5$<span class="inline-marks">[1]</span></div></li><li data-list-item-id="h2-2025-ejc-promo-q5-ejc-q5-item-block-2-item-1-sub-1" data-list-item-path="h2-2025-ejc-promo-q5-ejc-q5-item-block-2-item-1/h2-2025-ejc-promo-q5-ejc-q5-item-block-2-item-1-sub-1"><div class="list-item-paragraph">$x_1=1.5$<span class="inline-marks">[1]</span></div></li><li data-list-item-id="h2-2025-ejc-promo-q5-ejc-q5-item-block-2-item-1-sub-2" data-list-item-path="h2-2025-ejc-promo-q5-ejc-q5-item-block-2-item-1/h2-2025-ejc-promo-q5-ejc-q5-item-block-2-item-1-sub-2"><div class="list-item-paragraph">$x_1=-1$<span class="inline-marks">[1]</span></div></li></ol></li><li data-list-item-id="h2-2025-ejc-promo-q5-ejc-q5-item-block-2-item-2" data-list-item-path="h2-2025-ejc-promo-q5-ejc-q5-item-block-2-item-2"><div class="list-item-paragraph"><p>By considering the graph of $y=2x^2-x^3-x$, show that if&nbsp;$0&lt;x_n&lt;1$, then&nbsp;$x_{n+1}-x_n&lt;0$. State briefly what this result tells us about the sequence in part<strong> (b)(i)</strong>.</p><span class="inline-marks">[2]</span></div></li></ol>

The sequence $x_1, x_2, x_3, \ldots$ satisfies the recurrence relation \[x_{n+1}=x_n^2(2-x_n)\text{ for }n\geq 1\]

Prove algebraically that, if the sequence converges, then it converges to either 1 or 0.

Determine the behaviour of the sequence for each of the following cases.

2025 EJC Promo Q5

<p>A sequence $\{u_n\}$ of numbers are defined by $u_1 = 1$ and</p>
<p style="text-align: center">\[u_{n+1} = 2u_n + 4, \quad n \in \mathbb{N}.\]</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Find the values of $u_2, u_3$ and $u_4$.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Using GDC, find the least integral value of $n$ for which $u_n &gt; 256$.</span></div></li></ol>

A sequence $\{u_n\}$ of numbers are defined by $u_1 = 1$ and

\[u_{n+1} = 2u_n + 4, \quad n \in \mathbb{N}.\]

Find the values of $u_2, u_3$ and $u_4$.

Using GDC, find the least integral value of $n$ for which $u_n > 256$.

Recurrence relation: linear

<p>For each of the following, describe the behaviour of the sequence and determine whether the sequence is convergent or divergent. Find the limit of the sequence if it converges.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">$\left\{ \frac{1}{4}-\frac{2}{{{\left( 3n+1 \right)}^{2}}} \right\},n\in {{\mathbb{Z}}^{+}}$</div></li><li><div class="list-item-paragraph">$\left\{ \frac{5{{n}^{2}}+4n+1}{\left( 2n+1 \right)\left( 3n-2 \right)} \right\},n\in {{\mathbb{Z}}^{+}}$</div></li><li><div class="list-item-paragraph">$\left\{ {{\left( -2 \right)}^{n}} \right\},n\in {{\mathbb{Z}}^{+}}$</div></li><li><div class="list-item-paragraph">$\left\{ {{\left( \frac{1}{2} \right)}^{n}} \right\},n\in {{\mathbb{Z}}^{+}}$</div></li></ol>

For each of the following, describe the behaviour of the sequence and determine whether the sequence is convergent or divergent. Find the limit of the sequence if it converges.

NJC Sequence and Series Tutorial Q1

<p>A sequence of negative numbers is defined by ${{x}_{n+1}}=-\sqrt{8+3{{x}_{n}}}$ for $n=1$, $2$, $3$, …</p>
<ol class="i-list"><li><div class="list-item-paragraph">Show that $-\frac{8}{3}<{{x}_{n}}<0$ for $n=1$, $2$, $3$, …</div></li><li><div class="list-item-paragraph">Find the exact value of $l$, given that when $n\to \infty $, ${{x}_{n}}\to l$.</div></li><li><div class="list-item-paragraph">Show that ${{x}_{n+1}}^{2}-{{l}^{2}}=3\left( {{x}_{n}}-l \right)$.</div></li><li><div class="list-item-paragraph">Show algebraically that if ${{x}_{n}} \lt l $ for some value of $n$, then ${{x}_{n+1}} \gt l$.</div></li></ol>

A sequence of negative numbers is defined by ${{x}_{n+1}}=-\sqrt{8+3{{x}_{n}}}$ for $n=1$, $2$, $3$, …

Limit of a Sequence and Behaviour of a Sequence Q1

<p>The positive numbers ${{x}_{n}}$ satisfy the relation ${{x}_{n+1}}=\sqrt{{{x}_{n}}+5}$ for&nbsp;$n=1$, $2$, $3$, …</p>
<p>As&nbsp;$n\to \infty $, ${{x}_{n}}\to l$.</p>
<ol class="i-list"><li data-list-item-id="item-1760416493617-bujlx7khq" data-list-item-path="item-1760416493617-bujlx7khq"><div class="list-item-paragraph">Find (in either order) the value of $l$ to $3$ decimal places and the exact value of $l$.</div></li><li data-list-item-id="item-1760416494188-nqjdxn7os" data-list-item-path="item-1760416494188-nqjdxn7os"><div class="list-item-paragraph">Prove that&nbsp;${{\left( {{x}_{n+1}} \right)}^{2}}-{{l}^{2}}={{x}_{n}}-l$.</div></li><li data-list-item-id="item-1760416495485-f8wpo2in9" data-list-item-path="item-1760416495485-f8wpo2in9"><div class="list-item-paragraph">Hence show that, if&nbsp;${{x}_{n}}&gt;l$, then&nbsp;${{x}_{n}}&gt;{{x}_{n+1}}&gt;l$.</div></li></ol>

The positive numbers ${{x}_{n}}$ satisfy the relation ${{x}_{n+1}}=\sqrt{{{x}_{n}}+5}$ for $n=1$, $2$, $3$, …

2007 Specimen Paper P1 Q8

<p>The positive numbers ${{x}_{n}}$ satisfy the equation</p>
<p style="text-align: center">${{x}_{n+1}}=\sqrt{{{x}_{n}}+3}$, for&nbsp;$n=1$, $2$, $3$, …</p>
<p>As&nbsp;$n\to \infty $, ${{x}_{n}}\to \alpha $.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Find the exact value of $\alpha $.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Prove that&nbsp;${{\left( {{x}_{n+1}} \right)}^{2}}-{{\alpha }^{2}}={{x}_{n}}-\alpha $.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Hence, or otherwise, show that if&nbsp;${{x}_{n}}&gt;\alpha $ for a certain value of $n$, then&nbsp;${{x}_{n+2}}&gt;\alpha $.<span class="inline-marks">[3]</span></div></li></ol>

The positive numbers ${{x}_{n}}$ satisfy the equation

${{x}_{n+1}}=\sqrt{{{x}_{n}}+3}$, for $n=1$, $2$, $3$, …

As $n\to \infty $, ${{x}_{n}}\to \alpha $.

2007 VJC P1 Q6

<figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/converted/1760262586868-yy007nsbx8g.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NjAyNjI1ODY4NjgteXkwMDduc2J4OGcucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MDk2NzQ5MSwiZXhwIjoxNzkwOTY3NzkxfQ.Bfrf5FhP1F9dQz4asneFzalUzIg3ll3dCa0bLbeJdKQ" width="400" class="aligncenter" /></figure>
<p>The diagram shows the graph of $y=\ln \left( x+2 \right)-\frac{x}{5}$.</p>
<p>The positive root of the equation $\ln \left( x+2 \right)-\frac{x}{5}=0$ is denoted by $\alpha $.</p>
<p>Find the value of $\alpha $.</p>
<p>A sequence of positive real numbers ${{x}_{1}}$, ${{x}_{2}}$, ${{x}_{3}}$, … satisfies the recurrence relation</p>
<p style="text-align: center">${{x}_{n+1}}=5\ln \left( {{x}_{n}}+2 \right)$ for $n\ge 1$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Prove algebraically that, if the sequence converges, then it converges to $\alpha $.</div></li><li><div class="list-item-paragraph">Use a calculator to determine the behaviour of the sequence for each of the cases ${{x}_{1}}=1$ and ${{x}_{1}}=20$.</div></li><li><div class="list-item-paragraph">By considering ${{x}_{n+1}}-{{x}_{n}}$, prove that</div><div class="list-item-paragraph" style="text-align: center">${{x}_{n+1}}>{{x}_{n}}$ if $0<{{x}_{n}}<\alpha $</div><div class="list-item-paragraph" style="text-align: center">${{x}_{n+1}}<{{x}_{n}}$ if ${{x}_{n}}>\alpha $</div></li></ol>

The diagram shows the graph of $y=\ln \left( x+2 \right)-\frac{x}{5}$.

The positive root of the equation $\ln \left( x+2 \right)-\frac{x}{5}=0$ is denoted by $\alpha $.

A sequence of positive real numbers ${{x}_{1}}$, ${{x}_{2}}$, ${{x}_{3}}$, … satisfies the recurrence relation

${{x}_{n+1}}=5\ln \left( {{x}_{n}}+2 \right)$ for $n\ge 1$.

2018 IJC MYE Q12 [Modified]

<ol class="abc-list"><li><div class="list-item-paragraph">The volume of water, in litres, in a storage tank decreases by $10\%$ each day. However, $90$ litres of water is also pumped into the tank at the end of each day. The volume of water in the tank at the end of $n$ days is denoted by ${{x}_{n}}$ and ${{x}_{0}}$ is the initial volume of water in the tank.</div><ol class="i-list"><li><div class="list-item-paragraph">Write down a recurrence relation to represent the above situation.</div><p style="text-align: right;">[1]</p><div class="list-item-paragraph">Given that $a{{r}^{n-1}}+...+a{{r}^{2}}+ar+a=a\left( \frac{1-r{{\,}^{n}}}{1-r} \right)$,</div></li><li><div class="list-item-paragraph">Show that ${{x}_{n}}={{0.9}^{n}}\left( {{x}_{0}}-900 \right)+900$.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Deduce the limit of ${{x}_{n}}$ as $n$ becomes very large.</div><p style="text-align: right;">[1]</p></li></ol></li></ol>

2008 MJC P1 Q8

<p>A sequence ${{u}_{0}}$, ${{u}_{1}}$, ${{u}_{2}}$… is such that&nbsp;${{u}_{r}}=\frac{1}{r!}$ and ${{u}_{r}}={{u}_{r-2}}+\frac{r+1-{{r}^{2}}}{r!}$, when&nbsp;$n\ge 2$.</p>
<ol class="i-list"><li data-list-item-id="item-1759737554530-cznhathnh" data-list-item-path="item-1759737554530-cznhathnh"><div class="list-item-paragraph">Show that&nbsp;$\sum\limits_{r=2}^{n}{\frac{{{r}^{2}}-r-1}{r!}}=2-\frac{n+1}{n!}$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1759737555385-gdricbxr9" data-list-item-path="item-1759737555385-gdricbxr9"><div class="list-item-paragraph">Hence find $\sum\limits_{r=8}^{n+5}{\frac{{{r}^{2}}-3r+1}{\left( r-1 \right)!}}$ in terms of $n$.<span class="inline-marks">[3]</span></div></li></ol>
<p>Limit Comparison test states that for two series of the form $\sum\limits_{r=k}^{n}{{{a}_{r}}}$ and $\sum\limits_{r=k}^{n}{{{b}_{r}}}$ with ${{a}_{n}},{{b}_{n}}\ge 0$ for all $n$, if&nbsp;$\mathop{\lim}\limits_{n\to \infty }\,\frac{{{a}_{n}}}{{{b}_{n}}}&gt;0$, then both&nbsp;$\sum\limits_{r=k}^{\infty }{{{a}_{r}}}$and $\sum\limits_{r=k}^{\infty }{{{b}_{r}}}$ converges or both diverges.</p>
<ol class="i-list" start="3"><li data-list-item-id="item-1759737581942-t71zjddfx" data-list-item-path="item-1759737581942-t71zjddfx"><div class="list-item-paragraph">Given that&nbsp;$\sum\limits_{r=2}^{\infty }{\frac{{{r}^{2}}-r-1}{r!}}$ is convergent, using the test, explain why $\sum\limits_{r=2}^{\infty }{\frac{r-2}{\left( r-1 \right)!}}$ is convergent.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1759737582662-woqdbtvjv" data-list-item-path="item-1759737582662-woqdbtvjv"><div class="list-item-paragraph">Show that&nbsp;$e-2&lt;\frac{1}{2!}+\frac{2}{3!}+\frac{3}{4!}+\frac{4}{5!}+...&lt;2$.<span class="inline-marks">[3]</span></div></li></ol>

A sequence ${{u}_{0}}$, ${{u}_{1}}$, ${{u}_{2}}$… is such that ${{u}_{r}}=\frac{1}{r!}$ and ${{u}_{r}}={{u}_{r-2}}+\frac{r+1-{{r}^{2}}}{r!}$, when $n\ge 2$.

Limit Comparison test states that for two series of the form $\sum\limits_{r=k}^{n}{{{a}_{r}}}$ and $\sum\limits_{r=k}^{n}{{{b}_{r}}}$ with ${{a}_{n}},{{b}_{n}}\ge 0$ for all $n$, if $\underset{n\to \infty }{\mathop{\lim }}\,\frac{{{a}_{n}}}{{{b}_{n}}}>0$, then both $\sum\limits_{r=k}^{\infty }{{{a}_{r}}}$and $\sum\limits_{r=k}^{\infty }{{{b}_{r}}}$ converges or both diverges.

Manipulating Summation Limits

2018 SRJC P1 Q8

<p>The sum, ${{S}_{n}}$, of the first $n$ terms of a sequence ${{u}_{1}}$, ${{u}_{2}}$, ${{u}_{3}}$ ... is given by</p>
<p style="text-align: center">${{S}_{n}}=\ln \left( n+1 \right)$</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Find ${{u}_{n}}$ in terms of $n$.</span></div><p style="text-align: right;">[2]</p></li><li><div class="list-item-content"><span class="list-item-text">Describe the behaviour of the sequence.</span></div><p style="text-align: right;">[1]</p></li></ol>

The sum, ${{S}_{n}}$, of the first $n$ terms of a sequence ${{u}_{1}}$, ${{u}_{2}}$, ${{u}_{3}}$ ... is given by

Math Question Bank