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

<ol class="abc-list"><li data-list-item-id="imported-item-block-0-item-0" data-list-item-path="imported-item-block-0-item-0"><div class="list-item-paragraph"><p>The sum of the first $20$ terms of an arithmetic sequence is $65$. The sum of the next $8$ terms is $-30$.</p><p>Find the first term and the common difference of this sequence.</p><span class="inline-marks">[3]</span></div></li><li data-list-item-id="imported-item-block-0-item-1" data-list-item-path="imported-item-block-0-item-1"><div class="list-item-paragraph"><p>A sequence has&nbsp;$n$th term given by $u_n=e^{k(n+1)}$, where&nbsp;$k$ is a constant and $k\ne 0$.</p><p>Show that this sequence is geometric.</p><span class="inline-marks">[2]</span></div></li></ol>

Proving the progression

Block Sums

Arithmetic Progression

N2025 P1 Q2

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">In a geometric progression, the first term is 12 and the fourth term is $-\frac{3}{2}$. The sum of the first $n$ terms of the progression is given by $S_n$ and the sum to infinity of the progression is given by $S$. Find the least value of $n$ for which $\left| S - S_n \right| &lt; 0.001$.<span class="inline-marks">[5]</span></div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">An arithmetic progression has the first term 1000 and common difference $-1.5$. Calculate the value of the first negative term of the progression and the sum of all the positive terms.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">The sum of the first $n$ terms of a series is given by $S_n = \frac{1}{3}n(n + 12)$. By finding an expression for the $n$$^{th}$ term of the series, show that this is an arithmetic series. Hence, state the common difference of the arithmetic series.<span class="inline-marks">[4]</span></div></li></ol>

Problems involving inequalities

Sum to infinity

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

2024 MI PRU2 P1 Q12

<p>The sum, $S_n$, of the first $n$ terms of the sequence $u_1, u_2, u_3, \dots$ is given by</p>
<p style="text-align: center">\[S_n = n^2 + kn,\]</p>
<p>where&nbsp;$k$ is a non-zero constant.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Find $u_n$ in terms of $n$ and $k$.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Hence determine if the sequence is an arithmetic progression.<span class="inline-marks">[2]</span></div></li></ol>

The sum, $S_n$, of the first $n$ terms of the sequence $u_1, u_2, u_3, \dots$ is given by

2024 MI PRU1 P1 Q1

<ol class="abc-list"><li><div class="list-item-paragraph">In a geometric progression, the first term is $12$ and the fourth term is $-\frac{3}{2}$. The sum of the first $n$ terms of the 	progression is given by ${{S}_{n}}$ and the sum to infinity of the progression is given by $S$. Find the least value of $n$ for 	which $\left| S-{{S}_{n}} \right|&lt;0.001$. 	<span class="inline-marks">[5]</span></div></li><li><div class="list-item-paragraph">An arithmetic progression has the first term $1000$ and common difference $-1.5$. Calculate the value of the first 	negative term of the progression and the sum of all the positive terms. 	<span class="inline-marks">[4]</span></div></li><li><div class="list-item-paragraph">The sum of the first $n$ terms of a series is given by ${{S}_{n}}=\frac{1}{3}n\left( n+12 \right)$. By finding an expression for the ${{n}^{\text{th}}}$ term of the 	series, show that this is an arithmetic series. Hence, state the common difference of the arithmetic series.  <span class="inline-marks">[4]</span></div></li></ol>

<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">A sequence is such that&nbsp;$u_1 = a$ and $u_{n+1} = bu_n$, for&nbsp;$n \geq 1$. Both&nbsp;$a$ and $b$ are constants.</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">Explain why the sequence is a geometric progression.<span class="inline-marks">[1]</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">Given that&nbsp;$u_2 = 6$, $u_1 + u_2 + u_3 = 26$, and the sum to infinity exists, find the values of $a$ and $b$.<span class="inline-marks">[4]</span></div></li></ol></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph"><p>It is given that&nbsp;$\sum_{r=1}^{n} r^2 = \frac{1}{6}n(n+1)(2n+1).$</p></div><ol class="i-list"><li data-list-item-id="item-b-i" data-list-item-path="item-b/item-b-i"><div class="list-item-paragraph">Find $\sum_{r=1}^{n} r(r+2)$ in terms of $n$. Leave your answer in factorised form.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-b-ii" data-list-item-path="item-b/item-b-ii"><div class="list-item-paragraph">Hence find $\sum_{r=2}^{n} (r+1)(r+3)$ in terms of $n$. You do not need to leave your answer in factorised form.<span class="inline-marks">[3]</span></div></li></ol></li></ol>

A sequence is such that $u_1 = a$ and $u_{n+1} = bu_n$, for $n \geq 1$. Both $a$ and $b$ are constants.

Sigma Notation By Substitution

Manipulating Summation Limits

Geometric Progression

Summing Polynomial Series

2025 YIJC P2 Q4

<p>The sequence $\{u_n\}$ is a geometric sequence with first term $a$ and common ratio $\frac{3}{2}$, where&nbsp;$a &gt; 0$.</p>
<p>Another sequence $\{v_n\}$ is defined by $v_n = \frac{9}{u_n}$ for all positive integers $n$.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Show that&nbsp;$\{v_n\}$ is also a geometric sequence.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Given that&nbsp;$v_2 = u_2$, find the value of $a$.</span><span class="inline-marks">[2]</span></div></li></ol>

The sequence $\{u_n\}$ is a geometric sequence with first term $a$ and common ratio $\frac{3}{2}$, where $a > 0$.

Another sequence $\{v_n\}$ is defined by $v_n = \frac{9}{u_n}$ for all positive integers $n$.

Show that $\{v_n\}$ is also a geometric sequence.

Given that $v_2 = u_2$, find the value of $a$.

Inter-sequence Relationships

2025 TJC P1 Q1

<p>A series is given by $\sum_{r=1}^{n} 2(4-3x)^r$ where&nbsp;$x$ is constant.</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Explain why this is a geometric series. Determine the range of values of $x$ for the sum to infinity of this series to exist.<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">Using $x=1$, and given that&nbsp;$\sum_{r=1}^{n} r^2 = \frac{n}{6}(n+1)(2n+1)$, find $\sum_{r=0}^{n-1} [2(4-3x)^r + (r+1)(2r+5)]$, leaving your answer in the form of $n(an^2+bn+c)$, where&nbsp;$a$, $b$ and $c$ are constants to be determined.<span class="inline-marks">[4]</span></div></li></ol>

A series is given by $\sum_{r=1}^{n} 2(4-3x)^r$ where $x$ is constant.

Explain why this is a geometric series. Determine the range of values of $x$ for the sum to infinity of this series to exist.

Using $x=1$, and given that $\sum_{r=1}^{n} r^2 = \frac{n}{6}(n+1)(2n+1)$, find $\sum_{r=0}^{n-1} [2(4-3x)^r + (r+1)(2r+5)]$, leaving your answer in the form of $n(an^2+bn+c)$, where $a$, $b$ and $c$ are constants to be determined.

2025 DHS P1 Q2 

<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">The ninth, fifth and second terms of an arithmetic progression are successive terms of a geometric progression with first term $a$ and common ratio $r$, where&nbsp;$r \neq 1$ and $a &gt; 0$.</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">Find the value of $r$ and deduce that the geometric series is convergent.<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"><p>Using the value of $r$ found in<strong> (i)</strong>, find the least value of $n$ such that the sum of all the terms after the $n^{\mathrm{th}}$ term of the geometric progression is less than $1$% of its sum of the first $n$ terms.</p><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 sum, $S_n$, of the first $n$ terms of another sequence is given by $S_n = n \ln 2^{q(n+1)}$, where&nbsp;$q$ is a constant. Prove that the sequence follows an arithmetic progression.<span class="inline-marks">[4]</span></div></li></ol>

The ninth, fifth and second terms of an arithmetic progression are successive terms of a geometric progression with first term $a$ and common ratio $r$, where $r \neq 1$ and $a > 0$.

The sum, $S_n$, of the first $n$ terms of another sequence is given by $S_n = n \ln 2^{q(n+1)}$, where $q$ is a constant. Prove that the sequence follows an arithmetic progression.

2025 ASRJC P1 Q8

<p>The sum of the first $n$ terms of a sequence is given by $S_n = 4 - \frac{(-1)^n}{4^{n-1}}$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Find an expression for the $n$ th term of the sequence and show that the sequence is a geometric progression.</div></li><li><div class="list-item-content"><span class="list-item-text">Find the sum to infinity, $S$, of the progression and the least value of $n$ for which the magnitude of the difference between $S_n$ and $S$ is less than $0.001$.</span></div></li></ol>

The sum of the first $n$ terms of a sequence is given by $S_n = 4 - \frac{(-1)^n}{4^{n-1}}$.

Find the sum to infinity, $S$, of the progression and the least value of $n$ for which the magnitude of the difference between $S_n$ and $S$ is less than $0.001$.

2007 NJC Promo Q7

<p>The sum of the first $n$ terms of a sequence is given by ${{S}_{n}}=2{{n}^{2}}+n$. Prove that the series in an arithmetic series and state the common difference.</p>

The sum of the first $n$ terms of a sequence is given by ${{S}_{n}}=2{{n}^{2}}+n$. Prove that the series in an arithmetic series and state the common difference.

Worked Example 1

<p>If $a$ and $r$ are both positive, prove that the series</p>
<p style="text-align: center">$\log a + \log ar + \log ar^{2} + \log ar^{3} + ...+ \log ar^{n-1}$ </p>
<p>is an arithmetic series and find the sum.</p>

If $a$ and $r$ are both positive, prove that the series

$\log a + \log ar + \log ar^{2} + \log ar^{3} + ...+ \log ar^{n-1}$ 

is an arithmetic series and find the sum.

<p>The sum of the first $n$ terms of a sequence is given by $S_{n} = 5^{n} - 1$. State the first term and prove that the series is a geometric series. Thus, state the common ratio.</p>

The sum of the first $n$ terms of a sequence is given by $S_{n} = 5^{n} - 1$. State the first term and prove that the series is a geometric series. Thus, state the common ratio.

<p>The sum of the first $n$ terms of a sequence is given by $S_{n} = 6 - \dfrac{3^{n+1}}{2^{n-1}}$. State the first term and prove that the series is a geometric series. Thus, state the common ratio.</p>

The sum of the first $n$ terms of a sequence is given by $S_{n} = 6 - \dfrac{3^{n+1}}{2^{n-1}}$. State the first term and prove that the series is a geometric series. Thus, state the common ratio.

<p>The sum of the first $n$ terms of a sequence is given by $S_{n} = 3n - n^{2}$. Prove that the series is an arithmetic series and state the common difference.</p>

The sum of the first $n$ terms of a sequence is given by $S_{n} = 3n - n^{2}$. Prove that the series is an arithmetic series and state the common difference.

<p>The sum of the first $n$ terms of a sequence is given by $S_{n} = 2n^{2} + n$. Prove that the series is an arithmetic series and state the common difference.</p>

The sum of the first $n$ terms of a sequence is given by $S_{n} = 2n^{2} + n$. Prove that the series is an arithmetic series and state the common difference.

<p>A geometric sequence ${{u}_{1}}$, ${{u}_{2}}$, ${{u}_{3}}$, … has a common ratio $r$, where $r$ and ${{u}_{1}}$ are positive.</p>
<p>A sequence ${{v}_{1}}$, ${{v}_{2}}$,${{v}_{3}}$, … is given by ${{v}_{n}}=\ln \left( {{u}_{n}} \right)$ for all positive integer $n$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Show that ${{v}_{1}}$, ${{v}_{2}}$,${{v}_{3}}$, … is an arithmetic sequence.</div><p style="text-align: right;">[2]</p></li></ol>
<p>The sequence ${{u}_{1}}$, ${{u}_{2}}$, ${{u}_{3}}$, … is convergent.</p>
<ol class="abc-list" start="2"><li><div class="list-item-paragraph">Find the range of values of the common difference of the sequence ${{v}_{1}}$, ${{v}_{2}}$,${{v}_{3}}$, …</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Given that the sum to infinity of the sequence ${{u}_{1}}$, ${{u}_{2}}$, ${{u}_{3}}$, … is $9$ and ${{v}_{3}}=0$, find the possible values of $r$ and ${{u}_{1}}$.</div><p style="text-align: right;">[4]</p></li></ol>

A geometric sequence ${{u}_{1}}$, ${{u}_{2}}$, ${{u}_{3}}$, … has a common ratio $r$, where $r$ and ${{u}_{1}}$ are positive.

A sequence ${{v}_{1}}$, ${{v}_{2}}$,${{v}_{3}}$, … is given by ${{v}_{n}}=\ln \left( {{u}_{n}} \right)$ for all positive integer $n$.

The sequence ${{u}_{1}}$, ${{u}_{2}}$, ${{u}_{3}}$, … is convergent.

Limit of a Sequence and Behaviour of a Sequence

2022 EJC Promo Q8

<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.

Generating Terms from Recurrence Relations

Subsequences of Progressions

2024 EJC Promo Q11

<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>

Reverse Iteration of Recurrence Relations

2025 VJC P1 Q4

<ol class="abc-list"><li><div class="list-item-paragraph">The sum, ${{S}_{n}}$, of the first $n$ terms of a series is given by ${{S}_{n}}={{n}^{2}}+2n$. Show that this is an arithmetic series. Find the values of the first term and the common difference.<span class="inline-marks">[4]</span></div></li><li><div class="list-item-paragraph">In a geometric progression, the first term is $12$ and the sixth term is $-\frac{3}{8}$. Let the sum of the first $n$ terms of the progression and the sum to infinity be ${{S}_{n}}$ and $S$ respectively. Find the least value of $n$ for which the difference between ${{S}_{n}}$ and $S$ is less than $0.001$.)<span class="inline-marks">[4]</span></div></li></ol>

2024 NYJC Promo Q6

<p>The sum, ${{S}_{n}}$, of the first $n$ terms of the sequence ${{u}_{1}}$, ${{u}_{2}}$, ${{u}_{3}}$, $\ldots $ is given by</p>
<p style="text-align: center">${{S}_{n}}={{n}^{2}}+kn$,</p>
<p>where&nbsp;$k$ is a non-zero constant.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Find ${{u}_{n}}$ in terms of $n$ and $k$.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Hence determine if the sequence is an arithmetic progression.<span class="inline-marks">[2]</span></div></li></ol>

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

Math Question Bank