A Math

A Math (IGCSE)

E Math

E Math (IGCSE)

H1 Math

H2 Math

IB AA HL Math

IB AA SL Math

IB AI HL Math

IB AI SL Math

IP Math

N(A) A Math

N(A) E Math

PSLE

PSLE Foundation Mathematics

<p>A sequence $u_1, u_2, u_3, \ldots$ is defined by</p>
<p style="margin-left: 4rem">$u_1=a,$<br>$u_{r+1}=3u_r+5,  r\geqslant 1,$</p>
<p>where&nbsp;$a$ is a constant.</p>
<ol class="abc-list"><li data-list-item-id="imported-item-block-3-item-0" data-list-item-path="imported-item-block-3-item-0"><div class="list-item-paragraph">Find expressions for&nbsp;$u_2$ and $u_3$ in terms of $a$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="imported-item-block-3-item-1" data-list-item-path="imported-item-block-3-item-1"><div class="list-item-paragraph">It is given that&nbsp;$\displaystyle\sum_{r=1}^{4}u_r=110$. Find the value of $a$.<span class="inline-marks">[3]</span></div></li></ol>

A sequence $u_1, u_2, u_3, \ldots$ is defined by

$u_1=a,$
$u_{r+1}=3u_r+5,  r\geqslant 1,$

Find expressions for $u_2$ and $u_3$ in terms of $a$.

It is given that $\displaystyle\sum_{r=1}^{4}u_r=110$. Find the value of $a$.

Generating Terms from Recurrence Relations

N2025 P1 Q1

Sequence and Series

<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">For this question, you may use these results:</div><div class="list-item-paragraph" style="text-align: center">\[\sum_{r=1}^{n}r^2 = \frac{n(n+1)(2n+1)}{6} \quad \text{and} \quad \sum_{r=1}^{n}r^3 = \frac{n^2 (n+1)^2}{4}.\]</div><ol class="i-list"><li data-list-item-id="item-a-i" data-list-item-path="item-a/item-a-i"><div class="list-item-paragraph">Show that&nbsp;$\sum_{r=1}^{n}r(r+1)^2 = \frac{n(n+1)(n+2)(3n+5)}{12}$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-a-ii" data-list-item-path="item-a/item-a-ii"><div class="list-item-paragraph">Hence find $\sum_{r=5}^{n-1}(r+2)(r+3)^2$ in terms of $n$.<span class="inline-marks">[3]</span></div></li></ol></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">The sequence $u_1, u_2, u_3, \dots$ is defined by $u_1 = 2$, $u_{n+1} = \frac{1}{1-u_n}$, $n \ge 1$.</div><div class="list-item-paragraph">Find the values of $u_2$, $u_3$ and $u_4$. Hence find the value of $u_{2025}$.<span class="inline-marks">[3]</span></div></li></ol>

Sigma Notation By Substitution

Limit of a Sequence and Behaviour of a Sequence

Manipulating Summation Limits

Summing Polynomial Series

2025 ACJC P2 Q3

<p>A sequence $u_1, u_2, u_3, \dots$ is given by</p>
<p style="text-align: center">\[u_1 = 2 \text{ and}\hspace{0.5em} u_n = \frac{3u_{n-1} - 1}{6} \text{ for}\hspace{0.5em} n \ge 2.\]</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Find the exact values of $u_2$ and $u_3$.</span></div></li><li><div class="list-item-content"><span class="list-item-text">It is given that&nbsp;$u_n \to l$ as&nbsp;$n \to \infty$. Showing your working, find the exact value of $l$.</span></div></li><li><div class="list-item-paragraph">For this value of $l$, show that&nbsp;$u_n = \frac{14}{3}\left(\frac{1}{2}\right)^n + l$ satisfies both the initial condition and the recurrence relation given in the question.</div></li></ol>

A sequence $u_1, u_2, u_3, \dots$ is given by

\[u_1 = 2 \text{ and } u_n = \frac{3u_{n-1} - 1}{6} \text{ for } n \ge 2.\]

Find the exact values of $u_2$ and $u_3$.

It is given that $u_n \to l$ as $n \to \infty$. Showing your working, find the exact value of $l$.

Finding Explicit Terms of Linear Recurrences

N2012 P1 Q3 [Modified]

<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 $u_n > 256$.

Problems involving inequalities

Recurrence relation: linear

<p>A sequence is defined by $T_{n+2} = T_{n+1} + T_n$ with $T_1 = 3$, $T_2 = 1$ for&nbsp;$n \ge 1$. Another sequence is defined by $r_n = \frac{T_{n+1}}{T_n}$ for&nbsp;$n \ge 1$.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find $T_4$.</span><span class="inline-marks">[1]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Show that&nbsp;$r_{n+1} = 1 + \frac{1}{r_n}$ for all for&nbsp;$n \ge 1$.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">It is given that&nbsp;$r_n$ converges to a limit $L$. Find the exact value of $L$.</span><span class="inline-marks">[3]</span></div></li></ol>
<p>A third sequence $u_n$, for&nbsp;$n \ge 1$, is an arithmetic sequence with first term $5$ and common difference $2$.</p>
<ol class="abc-list" start="4"><li><div class="list-item-paragraph">Find the least value of $n$ such that&nbsp;$\displaystyle \sum_{k=1}^n u_k &gt; 930$.<span class="inline-marks">[2]</span></div></li></ol>

A sequence is defined by $T_{n+2} = T_{n+1} + T_n$ with $T_1 = 3$, $T_2 = 1$ for $n \ge 1$. Another sequence is defined by $r_n = \frac{T_{n+1}}{T_n}$ for $n \ge 1$.

Show that $r_{n+1} = 1 + \frac{1}{r_n}$ for all for $n \ge 1$.

It is given that $r_n$ converges to a limit $L$. Find the exact value of $L$.

A third sequence $u_n$, for $n \ge 1$, is an arithmetic sequence with first term $5$ and common difference $2$.

Arithmetic Progression

Inter-sequence Relationships

2025 MI P2 Q4

<p>In the $1^{st}$ month, a farm has a pair of rabbits. Then in the $3^{rd}$ month, the pair of rabbits grow up and reproduce another pair of rabbits. Suppose that: </p>
<p>1. It always takes two months for rabbits to grow up and be able to reproduce baby rabbits. </p>
<p>2. After growing up, every pair of rabbits reproduce one pair of rabbits every month. </p>
<p>3. No rabbit dies on this farm.</p>
<ol class="i-list"><li data-list-item-id="item-i" data-list-item-path="item-i"><div class="list-item-paragraph">Find the number of pair(s) of rabbits on the farm with respect to time (in month), by completing the following table. (Hint: from the $3^{\text{rd}}$ month onwards, the number of pair(s) of rabbits in each month equals the sum of the number of pair(s) of rabbits in the previous month and the number of pair(s) of new-born rabbits this month.)</div></li></ol>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 400px;"><colgroup><col style="width: 37.5000%;"><col style="width: 62.5000%;"></colgroup><thead><tr><th style="text-align: center; --cell-justify: center">Time (Month)</th><th style="text-align: center; --cell-justify: center">Number of Pair(s) of Rabbits, $P_n$</th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center">$1$</td><td style="text-align: center; --cell-justify: center">$1$</td></tr><tr><td style="text-align: center; --cell-justify: center">$2$</td><td style="text-align: center; --cell-justify: center">$1$</td></tr><tr><td style="text-align: center; --cell-justify: center">$3$</td><td style="text-align: center; --cell-justify: center">$2$</td></tr><tr><td style="text-align: center; --cell-justify: center">$4$</td><td style="text-align: center; --cell-justify: center">$3$</td></tr><tr><td style="text-align: center; --cell-justify: center">$5$</td><td style="text-align: center; --cell-justify: center">$5$</td></tr><tr><td style="text-align: center; --cell-justify: center">$6$</td><td style="text-align: center; --cell-justify: center">$8$</td></tr><tr><td style="text-align: center; --cell-justify: center">$7$</td><td style="text-align: center; --cell-justify: center"></td></tr><tr><td style="text-align: center; --cell-justify: center">$8$</td><td style="text-align: center; --cell-justify: center"></td></tr><tr><td style="text-align: center; --cell-justify: center">$9$</td><td style="text-align: center; --cell-justify: center"></td></tr></tbody></table></div></div>
<ol class="i-list" start="2"><li data-list-item-id="item-ii" data-list-item-path="item-ii"><div class="list-item-paragraph">State a rule of the number sequence formed by the number of pair(s) of rabbits on the farm, $P_n$.</div></li><li data-list-item-id="item-iii" data-list-item-path="item-iii"><div class="list-item-paragraph">Based on your answers in <strong>(i)</strong> and <strong>(ii)</strong>, Write down the next two terms of the following sequences.</div><ol class="abc-list"><li data-list-item-id="item-iii-a" data-list-item-path="item-iii/item-iii-a"><div class="list-item-paragraph">$-3, 2, -1, 1, 0, \dots$</div></li><li data-list-item-id="item-iii-b" data-list-item-path="item-iii/item-iii-b"><div class="list-item-paragraph">$3, 2, 5, 7, 12, \dots$</div></li></ol></li></ol>

In the $1^{st}$ month, a farm has a pair of rabbits. Then in the $3^{rd}$ month, the pair of rabbits grow up and reproduce another pair of rabbits. Suppose that: 

1. It always takes two months for rabbits to grow up and be able to reproduce baby rabbits. 

2. After growing up, every pair of rabbits reproduce one pair of rabbits every month. 

Find the number of pair(s) of rabbits on the farm with respect to time (in month), by completing the following table. (Hint: from the $3^{\text{rd}}$ month onwards, the number of pair(s) of rabbits in each month equals the sum of the number of pair(s) of rabbits in the previous month and the number of pair(s) of new-born rabbits this month.)

State a rule of the number sequence formed by the number of pair(s) of rabbits on the farm, $P_n$.

Modelling with Recurrence Relations

 Number Patterns

<p>For each of the following recurrence relations, explain how to obtain the next term in the sequence based on the given formula, describe the relationship represented by the recurrence relation, and calculate and write down the first $5$ terms of the sequence starting from the initial value provided.</p>
<ol class="i-list"><li><div class="list-item-paragraph">${{x}_{n+1}}=\sqrt{{{x}_{n}}+5}$, ${{x}_{0}}=1$.</div></li><li><div class="list-item-paragraph">${{u}_{n+1}}=0.5{{u}_{n}}+8$, ${{u}_{1}}=4$.</div></li><li><div class="list-item-paragraph">${{a}_{n}}={{a}_{n-1}}+k$, ${{a}_{0}}=3$.</div></li><li><div class="list-item-paragraph">${{u}_{n}}=-2{{u}_{n-1}}$, ${{u}_{0}}=3$.</div></li><li><div class="list-item-paragraph">${{T}_{n}}={{T}_{n-1}}+{{T}_{n-2}}$, ${{T}_{1}}=0$, ${{T}_{2}}=1$.</div></li></ol>

For each of the following recurrence relations, explain how to obtain the next term in the sequence based on the given formula, describe the relationship represented by the recurrence relation, and calculate and write down the first $5$ terms of the sequence starting from the initial value provided.

Worked Example 4

<p>A sequence of real numbers, ${{x}_{n}}$, satisfies the recurrence relation</p>
<p style="text-align: center">${{x}_{n+1}}=a{{\left( -3 \right)}^{n}}+b{{n}^{2}}+c{{x}_{n}}$, for $n\in \mathbb{Z}$, $a\in \mathbb{Z}$, $b\in \mathbb{Z}$, $c\in \mathbb{Z}$ and $n\ge 1$.</p>
<p>Given that ${{x}_{1}}=1$, ${{x}_{2}}=-8$, ${{x}_{3}}=70$, and ${{x}_{4}}=-377$, find the values of $a$, $b$ and $c$.</p><p style="text-align: right;">[3]</p>
<p>Hence find the values of ${{x}_{10}}$ and ${{x}_{11}}$.</p><p style="text-align: right;">[2]</p>

A sequence of real numbers, ${{x}_{n}}$, satisfies the recurrence relation

${{x}_{n+1}}=a{{\left( -3 \right)}^{n}}+b{{n}^{2}}+c{{x}_{n}}$, for $n\in \mathbb{Z}$, $a\in \mathbb{Z}$, $b\in \mathbb{Z}$, $c\in \mathbb{Z}$ and $n\ge 1$.

Given that ${{x}_{1}}=1$, ${{x}_{2}}=-8$, ${{x}_{3}}=70$, and ${{x}_{4}}=-377$, find the values of $a$, $b$ and $c$.

Hence find the values of ${{x}_{10}}$ and ${{x}_{11}}$.

Determining Model Coefficients Using Linear Systems

2025 HCI P1 Q2

System of Linear Equations

<p>A sequence is given by:</p>
<p style="margin-left: 15.5rem">${x_1 = 1,}$<br>${x_{n+1} = x_n (p + x_n),}$</p>
<p>where&nbsp;$p$ is a constant ($p\ne 0$).</p>
<ol class="abc-list"><li data-list-item-id="item-1760870901910-m3qkjw764" data-list-item-path="item-1760870901910-m3qkjw764"><div class="list-item-paragraph">Find ${{x}_{2}}$ in terms of $p$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1760870902555-euf4gwhx5" data-list-item-path="item-1760870902555-euf4gwhx5"><div class="list-item-paragraph">Show that&nbsp;${{x}_{3}}=1+3p+2{{p}^{2}}$.<span class="inline-marks">[2]</span></div></li></ol>
<p>Given that&nbsp;${{x}_{3}}=1$,</p>
<ol class="abc-list" start="3"><li data-list-item-id="item-1760870916910-lrmsqtxxn" data-list-item-path="item-1760870916910-lrmsqtxxn"><div class="list-item-paragraph">find the value of $p$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1760870917833-x337v8zbe" data-list-item-path="item-1760870917833-x337v8zbe"><div class="list-item-paragraph">Write down the value of ${{x}_{2008}}$.<span class="inline-marks">[2]</span></div></li></ol>

${x_1 = 1,}$
${x_{n+1} = x_n (p + x_n),}$

2008 A-Level Edexcel C1 January Q7

<p>A sequence of numbers is defined by the recurrence relation for&nbsp;$n\ge 1$</p>
<p style="text-align: center">${{u}_{n+1}}=k{{u}_{n}}+6\quad {{u}_{1}}=4$,</p>
<p>where&nbsp;$k$ is a non zero constant.</p>
<ol class="abc-list"><li data-list-item-id="item-1760870812923-zzsh0r863" data-list-item-path="item-1760870812923-zzsh0r863"><div class="list-item-paragraph">Find, in terms of $k$, the value of ${{u}_{2}}$ and the value of ${{u}_{3}}$.</div></li><li data-list-item-id="item-1760870813590-gudwjgxor" data-list-item-path="item-1760870813590-gudwjgxor"><div class="list-item-paragraph">Given that&nbsp;${{u}_{3}}=10$ find the possible values of $k$.</div></li></ol>
<p>The sequence tends to a limit $L$.</p>
<ol class="abc-list" start="3"><li data-list-item-id="item-1760870824482-t8ezswyjs" data-list-item-path="item-1760870824482-t8ezswyjs"><div class="list-item-paragraph">Find the value of ${{u}_{4}}$.</div></li><li data-list-item-id="item-1760870825508-w6jaxrk88" data-list-item-path="item-1760870825508-w6jaxrk88"><div class="list-item-paragraph">Determine the value of $L$.</div></li></ol>

A sequence of numbers is defined by the recurrence relation for $n\ge 1$

${{u}_{n+1}}=k{{u}_{n}}+6\quad {{u}_{1}}=4$,

T.Madas - Recurrence Relations Question 41

<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 ${{t}_{5}}=200$.

Reverse Iteration of Recurrence Relations

T.Madas - Recurrence Relations Q37

<p>A recurrence relation is defined for&nbsp;$n\ge 1$ by</p>
<p style="text-align: center">${{t}_{n+1}}=k{{t}_{n}}-1,\quad {{t}_{1}}=2$,</p>
<p>where&nbsp;$k$ is a non-zero constant.</p>
<ol class="abc-list"><li data-list-item-id="item-1760870091407-njazklrfk" data-list-item-path="item-1760870091407-njazklrfk"><div class="list-item-paragraph">Find the value of ${{t}_{3}}$ in terms of $k$.</div></li><li data-list-item-id="item-1760870092195-bkrv9o1xb" data-list-item-path="item-1760870092195-bkrv9o1xb"><div class="list-item-paragraph">Given that&nbsp;${{t}_{3}}=14$ find the possible values of $k$.</div></li></ol>

A recurrence relation is defined for $n\ge 1$ by

${{t}_{n+1}}=k{{t}_{n}}-1,\quad {{t}_{1}}=2$,

T.Madas - Recurrence Relations Q8

<ol class="abc-list"><li><div class="list-item-paragraph">For this question, you may use these results:</div><div class="list-item-paragraph" style="text-align: center">$\sum\limits_{r=1}^{n}{{{r}^{2}}}=\frac{n(n+1)(2n+1)}{6}$ and $\sum\limits_{r=1}^{n}{{{r}^{3}}}=\frac{{{n}^{2}}{{(n+1)}^{2}}}{4}$.</div><ol class="i-list"><li><div class="list-item-paragraph">Show that $\sum\limits_{r=1}^{n}{r}{{(r+1)}^{2}}=\frac{n(n+1)(n+2)(3n+5)}{12}$.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Hence find $\sum\limits_{r=5}^{n-1}{(r+2)}{{(r+3)}^{2}}$ in terms of $n$.</div><p style="text-align: right;">[3]</p></li></ol></li><li><div class="list-item-paragraph">The sequence ${{u}_{1}},{{u}_{2}},{{u}_{3}},\ldots $ is defined by ${{u}_{1}}=2$, ${{u}_{n+1}}=\frac{1}{1-{{u}_{n}}}$, $n\ge 1$.</div><div class="list-item-paragraph">Find the values of ${{u}_{2}}$, ${{u}_{3}}$ and ${{u}_{4}}$. Hence find the value of ${{u}_{2025}}$.</div><p style="text-align: right;">[3]</p></li></ol>

<p>A sequence is defined by the recurrence relation</p>
<p style="text-align: center">${{u}_{n+1}}=\frac{1+{{u}_{n}}}{1-{{u}_{n}}}$,</p>
<p>for $n\ge 1$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">State what happens to the sequence when ${{u}_{1}}=0$.</div><p style="text-align: right;">[1]</p></li></ol>
<p>It is now given that ${{u}_{1}}=2$.</p>
<ol class="abc-list" start="2"><li><div class="list-item-paragraph">Find ${{u}_{2}},\ {{u}_{3}},\ {{u}_{4}},\ {{u}_{5}}$ and ${{u}_{6}}$.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">By observing the pattern in part <strong>(b)</strong>, find $\sum\limits_{r=1}^{4n}{{{u}_{r}}}$ in terms of $n$.</div><p style="text-align: right;">[2]</p></li></ol>

A sequence is defined by the recurrence relation

${{u}_{n+1}}=\frac{1+{{u}_{n}}}{1-{{u}_{n}}}$,

2024 HCI P1 Q2

<p>A sequence ${{u}_{1}}$, ${{u}_{2}}$, ${{u}_{3}}$, $\ldots $ is such that&nbsp;${{u}_{1}}=2$, ${{u}_{n+1}}=k{{u}_{n}}-3$ for all $n\ge 1$, where&nbsp;$k$ is a constant.</p>
<ol class="abc-list"><li data-list-item-id="item-1760867451046-l46tujqyk" data-list-item-path="item-1760867451046-l46tujqyk"><div class="list-item-paragraph">Let&nbsp;$k=\frac{1}{5}$.</div><ol class="i-list"><li data-list-item-id="subitem-1760867467899-nat0vhbk3" data-list-item-path="item-1760867451046-l46tujqyk/subitem-1760867467899-nat0vhbk3"><div class="list-item-paragraph">Write down the values of ${{u}_{2}}$, ${{u}_{3}}$ and ${{u}_{4}}$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="subitem-1760867468612-b7nq92c8p" data-list-item-path="item-1760867451046-l46tujqyk/subitem-1760867468612-b7nq92c8p"><div class="list-item-paragraph">Given that the sequence converges to $l$, find the value of $l$.<span class="inline-marks">[1]</span></div></li></ol></li><li data-list-item-id="item-1760867451901-w7wkrdudv" data-list-item-path="item-1760867451901-w7wkrdudv"><div class="list-item-paragraph">Let&nbsp;$k=1$.</div><ol class="i-list"><li data-list-item-id="subitem-1760867487451-webp25nu4" data-list-item-path="item-1760867451901-w7wkrdudv/subitem-1760867487451-webp25nu4"><div class="list-item-paragraph">Given that the sequence ${{u}_{1}}$, ${{u}_{2}}$, ${{u}_{3}}$, $\ldots $ is an arithmetic progression, find the value of $\sum\limits_{r=1}^{n}{{{u}_{r+4}}}$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="subitem-1760867488366-myko8gzzd" data-list-item-path="item-1760867451901-w7wkrdudv/subitem-1760867488366-myko8gzzd"><div class="list-item-paragraph">Show that the sequence ${{u}_{7}}$, ${{u}_{10}}$, ${{u}_{13}}$, ${{u}_{16}}$, $\ldots $, ${{u}_{3n+4}}$, $\ldots $ is also an arithmetic progression.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="subitem-1760867488981-sr7j3o52q" data-list-item-path="item-1760867451901-w7wkrdudv/subitem-1760867488981-sr7j3o52q"><div class="list-item-paragraph">Hence, find the value of $\sum\limits_{r=1}^{100}{{{u}_{3r+4}}}$.<span class="inline-marks">[2]</span></div></li></ol></li></ol>

A sequence ${{u}_{1}}$, ${{u}_{2}}$, ${{u}_{3}}$, $\ldots $ is such that ${{u}_{1}}=2$, ${{u}_{n+1}}=k{{u}_{n}}-3$ for all $n\ge 1$, where $k$ is a constant.

Proving the progression

Block Sums

Subsequences of Progressions

2024 EJC Promo Q11

<ol class="abc-list"><li data-list-item-id="item-1760867251696-a8zdnrer1" data-list-item-path="item-1760867251696-a8zdnrer1"><div class="list-item-paragraph">It is given that&nbsp;$\sum\limits_{r=1}^{n}{\frac{1}{4{{r}^{2}}-1}}=\frac{n}{2n+1}$.</div><ol class="i-list"><li data-list-item-id="subitem-1760867257524-py2jezk0q" data-list-item-path="item-1760867251696-a8zdnrer1/subitem-1760867257524-py2jezk0q"><div class="list-item-paragraph">Explain why the series $\sum\limits_{r=1}^{\infty }{\frac{1}{4{{r}^{2}}-1}}$ converges and write down the value to which it converges.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="subitem-1760867261296-f0f2tstpv" data-list-item-path="item-1760867251696-a8zdnrer1/subitem-1760867261296-f0f2tstpv"><div class="list-item-paragraph">Find $\sum\limits_{r=6}^{N}{\frac{1}{\left( 2r+1 \right)\left( 2r+3 \right)}}$ in terms of $N$, express your answer in a single fraction.<span class="inline-marks">[3]</span></div></li></ol></li><li data-list-item-id="item-1760867252289-qap15ha8l" data-list-item-path="item-1760867252289-qap15ha8l"><div class="list-item-paragraph">The sequence ${{u}_{1}}$, ${{u}_{2}}$, ${{u}_{3}}$, $\ldots $ is defined by ${{u}_{1}}=2$, ${{u}_{n+1}}=1-\frac{1}{{{u}_{n}}}$, $n\ge 1$.</div><div class="list-item-paragraph">Find the value of ${{u}_{2}}$, ${{u}_{3}}$ and ${{u}_{4}}$. Hence find the value of $\sum\limits_{r=1}^{50}{{{u}_{r}}}$.<span class="inline-marks">[4]</span></div></li></ol>

Application of Sum to Infinity

2024 ACJC Promo Q8

<p>A sequence ${{u}_{1}},{{u}_{2}},{{u}_{3}},\ldots $ is such that ${{u}_{1}}=2$, ${{u}_{n+1}}=\frac{{{u}_{n}}}{5}-3$ for all $n\ge 1$, where $k$ is a constant.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Write down the values of ${{u}_{2}}$, ${{u}_{3}}$ and ${{u}_{4}}$.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Given that the sequence converges to $l$, find the value of $l$.</div><p style="text-align: right;">[1]</p></li></ol>

A sequence ${{u}_{1}},{{u}_{2}},{{u}_{3}},\ldots $ is such that ${{u}_{1}}=2$, ${{u}_{n+1}}=\frac{{{u}_{n}}}{5}-3$ for all $n\ge 1$, where $k$ is a constant.

RI Sequences and Series tutorial 7A Q4

<p>A number sequence with the first term, ${{T}_{1}}=1$, is defined by the following rule for any integer $n\ge 1$:</p>
<p>\[ T_{n+1} = 
\begin{cases} 
2T_n, & T_n \leq 7 \\
T_n - 7, & T_n &gt; 7 
\end{cases} \]</p>
<p>Find the sum of first $102$ terms of this number sequence.</p>

A number sequence with the first term, ${{T}_{1}}=1$, is defined by the following rule for any integer $n\ge 1$:

\[ T_{n+1} = 
\begin{cases} 
2T_n, & T_n \leq 7 \\
T_n - 7, & T_n > 7 
\end{cases} \]

Find the sum of first $102$ terms of this number sequence.

IP1U7 HOT Q9

<p>A sequence is defined by the recurrence relation</p>
<p style="text-align: center">${{u}_{n+1}}=\frac{1+{{u}_{n}}}{1-{{u}_{n}}}$,</p>
<p>for&nbsp;$n\ge 1$.</p>
<ol class="abc-list"><li data-list-item-id="item-1760690257136-gcyakuyx4"><div class="list-item-paragraph">State what happens to the sequence when&nbsp;${{u}_{1}}=0$.<span class="inline-marks">[1]</span></div></li></ol>
<p>It is now given that&nbsp;${{u}_{1}}=2$.</p>
<ol class="abc-list" start="2"><li data-list-item-id="item-1760690279411-cbeq9gt17"><div class="list-item-paragraph">Find ${{u}_{2}}$, ${{u}_{3}}$, ${{u}_{4}}$, ${{u}_{5}}$, and ${{u}_{6}}$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1760690280116-ht8mshoc4"><div class="list-item-paragraph">By observing the pattern in part <strong>(b)</strong>, find $\displaystyle \underset{r=1}{\overset{4n}{\mathop \sum }}\,{{u}_{r}}$ in terms of $n$.<span class="inline-marks">[2]</span></div></li></ol>

2024 HCI P2 Q2

<p>Consider the sequence defined by ${{u}_{n+1}}=\frac{2}{{{u}_{n}}-2}+3$, for all positive integers of $n$, and ${{u}_{0}}=3$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Find the values of ${{u}_{1}}$, ${{u}_{2}}$ and ${{u}_{3}}$.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Suppose ${{u}_{n}}$ converges to $l$ as $n\to \infty $. By forming and solving a suitable quadratic equation, find the value of $l$.</div><p style="text-align: right;">[4]</p></li><li><div class="list-item-paragraph">Under what condition will ${{u}_{n}}$ converge to the other limit you rejected in <strong>(ii)</strong>?</div><div class="list-item-paragraph">You should use calculations obtained from your G.C. to substantiate your answer.</div><p style="text-align: right;">[1]</p></li></ol>

Consider the sequence defined by ${{u}_{n+1}}=\frac{2}{{{u}_{n}}-2}+3$, for all positive integers of $n$, and ${{u}_{0}}=3$.

Math Question Bank