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N(A) E Math

PSLE

PSLE Foundation Mathematics

<p>An arithmetic series has first term $a$ and common difference $d$, where&nbsp;$a$ and $d$ are non-zero real constants. A convergent geometric series has first term $b$ and common ratio $r$, where&nbsp;$b &gt; 0$ and $r$ is a non-zero real constant. The sum of the first $n$ terms of the geometric series is denoted by $S$. It is given that the $7$th, $10$th and $11$th terms of the arithmetic series are equal to the $3$rd, $6$th and $10$th terms of the geometric series respectively.</p>
<ol class="abc-list"><li data-list-item-id="item-3a" data-list-item-path="item-3a"><div class="list-item-paragraph">Show that&nbsp;$r$ satisfies the equation $3r^7 - 4r^3 + 1 = 0$. Solve this equation, giving your answer correct to $4$ decimal places.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-3b" data-list-item-path="item-3b"><div class="list-item-paragraph">A new series is formed by taking the even numbered terms of the geometric series. Find the smallest value of $n$ for which $S$ differs from the sum to infinity of the new series by at least $1.78b$.<span class="inline-marks">[4]</span></div></li></ol>

An arithmetic series has first term $a$ and common difference $d$, where $a$ and $d$ are non-zero real constants. A convergent geometric series has first term $b$ and common ratio $r$, where $b > 0$ and $r$ is a non-zero real constant. The sum of the first $n$ terms of the geometric series is denoted by $S$. It is given that the $7$th, $10$th and $11$th terms of the arithmetic series are equal to the $3$rd, $6$th and $10$th terms of the geometric series respectively.

Show that $r$ satisfies the equation $3r^7 - 4r^3 + 1 = 0$. Solve this equation, giving your answer correct to $4$ decimal places.

A new series is formed by taking the even numbered terms of the geometric series. Find the smallest value of $n$ for which $S$ differs from the sum to infinity of the new series by at least $1.78b$.

Problems involving inequalities

Sum to infinity

Subsequences of Progressions

Inter-sequence Relationships

2025 TMJC P1 Q4

Sequence and Series

<p>A convergent geometric progression has a positive first term $a$ and common ratio $r$ while an arithmetic progression has a non-zero common difference $d$. The second and third term of the geometric progression is the first and fourth term of the arithmetic progression respectively. It is also given that the first term of the geometric progression is equal to the sum of the first $4$ odd-numbered terms in the arithmetic progression.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">By expressing the first term of the arithmetic progression in terms of $a$ and $r$, show that&nbsp;$r^2 = \frac{1}{4}$.</span><span class="inline-marks">[4]</span></div></li><li><div class="list-item-content"><span class="list-item-text">It is given that the sum to infinity of the geometric progression is less than $a$.</span></div><ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find the value of $r$, justifying your answer.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Explain if this implies that for all $n$, the sum of the first $n$ terms of the geometric progression must be less than $a$.</span><span class="inline-marks">[1]</span></div></li></ol></li></ol>

A convergent geometric progression has a positive first term $a$ and common ratio $r$ while an arithmetic progression has a non-zero common difference $d$. The second and third term of the geometric progression is the first and fourth term of the arithmetic progression respectively. It is also given that the first term of the geometric progression is equal to the sum of the first $4$ odd-numbered terms in the arithmetic progression.

By expressing the first term of the arithmetic progression in terms of $a$ and $r$, show that $r^2 = \frac{1}{4}$.

It is given that the sum to infinity of the geometric progression is less than $a$.

2013 NJC P1 Q4

<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

Limit of a Sequence and Behaviour of a Sequence

Generating Terms from Recurrence Relations

2024 EJC Promo Q11

<p>The fifth, tenth and twentieth term of a convergent geometric progression, ${G}$, are the first three consecutive terms of an arithmetic progression, ${A}$.</p>
<ol class="i-list"><li data-list-item-id="item-1760839937177-psbt8wu4o" data-list-item-path="item-1760839937177-psbt8wu4o"><div class="list-item-paragraph">Determine the common ratio ${G}$.<span class="inline-marks">[3]</span></div></li></ol>
<p>Given that the first term of ${G}$ is ${2}$.</p>
<ol class="i-list" start="2"><li data-list-item-id="item-1760839950954-k9l91lgsg" data-list-item-path="item-1760839950954-k9l91lgsg"><div class="list-item-paragraph">evaluate the sum to infinity for&nbsp;${G}$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1760839952185-xhheuhrbf" data-list-item-path="item-1760839952185-xhheuhrbf"><div class="list-item-paragraph">find the sum of the first ${10}$ odd-numbered terms in ${A}$.<span class="inline-marks">[3]</span></div></li></ol>

The fifth, tenth and twentieth term of a convergent geometric progression, ${G}$, are the first three consecutive terms of an arithmetic progression, ${A}$.

Given that the first term of ${G}$ is ${2}$.

<ol class="abc-list"><li><div class="list-item-paragraph">An arithmetic progression has first term $a$ and common difference $d$, where $a>0$ and $d\ne 0$. The eighth, third and second term of the progression are the first three terms of an infinite geometric progression.</div><ol class="i-list"><li><div class="list-item-paragraph">Find the common ratio of the geometric progression.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Find the exact sum of the odd-numbered terms of the geometric progression in terms of $a$.</div><p style="text-align: right;">[3]</p></li></ol></li><li><div class="list-item-paragraph">A programmer coded a program involving a rabbit-fox chase along a straight path to model the actual hunt for a rabbit by a fox. The rabbit first hop is $1.75$m. In each subsequent hop, the distance covered is $1\%$ less than its previous hop. The fox first leaps $3$ m. In each subsequent leap, the distance covered is $0.02$m less than its previous leap. Initially the rabbit is $60$ m ahead of the fox and assume that the rabbit and the fox start and end each hop and leap at the same time.</div><ol class="i-list"><li><div class="list-item-paragraph">By finding the total distance travelled by the fox and the rabbit after n leaps and hops respectively, find the minimum number of hops and leaps for the fox to catch up with the rabbit.</div><p style="text-align: right;">[4]</p></li><li><div class="list-item-paragraph">Find the number of leaps the fox takes before it comes to a stop. Hence, find the minimum starting distance, in metre, between the fox and the rabbit such that the fox will never catch up with the rabbit. Leave your answer to the nearest integer.</div><p style="text-align: right;">[2]</p></li></ol></li></ol>

Geometric Progression

Arithmetic Progression

2022 ASRJC P1 Q9

<ol class="abc-list"><li data-list-item-id="item-1760448674318-sum5tm3vr" data-list-item-path="item-1760448674318-sum5tm3vr"><ol class="i-list"><li data-list-item-id="subitem-1760448675390-ndyz8magz" data-list-item-path="item-1760448674318-sum5tm3vr/subitem-1760448675390-ndyz8magz"><div class="list-item-paragraph">The sum of the first $n$  terms of a sequence ${{u}_{1}}$, ${{u}_{2}}$, ${{u}_{3}}$, … is given by ${{S}_{n}}={{\mathrm{e}}^{2}}{{n}^{2}}-3n$.</div><div class="list-item-paragraph">Show that&nbsp;${{u}_{n}}={{\mathrm{e}}^{2}}\left( 2n-1 \right)-3$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="subitem-1760448676380-m8f6zot5h" data-list-item-path="item-1760448674318-sum5tm3vr/subitem-1760448676380-m8f6zot5h"><div class="list-item-paragraph">Hence show that the sequence is an arithmetic progression.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="subitem-1760448677265-vfnajhz3p" data-list-item-path="item-1760448674318-sum5tm3vr/subitem-1760448677265-vfnajhz3p"><div class="list-item-paragraph">State the first $3$ odd-numbered terms, ${{u}_{1}}$, ${{u}_{3}}$, and ${{u}_{5}}$. Hence find the least value of $n$ such that the sum of the first $n$ odd-numbered terms is more than $500$.<span class="inline-marks">[3]</span></div></li></ol></li><li data-list-item-id="item-1760448700028-hrb8cm3mg" data-list-item-path="item-1760448700028-hrb8cm3mg"><ol class="i-list"><li data-list-item-id="subitem-1760448701168-jt9wne8g9" data-list-item-path="item-1760448700028-hrb8cm3mg/subitem-1760448701168-jt9wne8g9"><div class="list-item-paragraph">Find the range of values of $x$ for which the following geometric series converges,</div><div class="list-item-paragraph" style="text-align: center">$\mathrm{e}-{{\mathrm{e}}^{\frac{1}{x}+1}}+{{\mathrm{e}}^{\frac{2}{x}+1}}-{{\mathrm{e}}^{\frac{3}{x}+1}}+...$<span class="inline-marks">[2]</span></div></li><li data-list-item-id="subitem-1760448702398-veol6qtyd" data-list-item-path="item-1760448700028-hrb8cm3mg/subitem-1760448702398-veol6qtyd"><div class="list-item-paragraph">Given that&nbsp;$x=-1$, find the sum of all the terms of this geometric series after, and including, the ${{8}^{\mathrm{th}}}$ term.<span class="inline-marks">[3]</span></div></li></ol></li></ol>

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

2023 ACJC Promo Q8

<p>The series $a+ar-a{{r}^{2}}-a{{r}^{3}}+a{{r}^{4}}+a{{r}^{5}}-a{{r}^{6}}-a{{r}^{7}}+$…, where $a&gt;0$, has its $k$th term, ${{T}_{k}}$, defined by ${{T}_{k}}=\left\{ \begin{matrix}
a{{r}^{k-1}}, \\
-a{{r}^{k-1}}, \\
\end{matrix} \right.$ $\begin{matrix}
\mathrm{if}\,k=4p-3\,\,\mathrm{or}\,\,4p-2 \\
\mathrm{if}\,k=4p-1\,\,\mathrm{or}\,\,4p\,\,\,\,\,\,\, \\
\end{matrix}$, for $p\in {{\mathbb{Z}}^{+}}$,</p>
<ol class="abc-list"><li><div class="list-item-paragraph">By rewriting the series as the sum of two geometric series, or otherwise, prove that the sum to $4n$ terms of the series is $\frac{a(1+r)(1-{{r}^{4n}})}{1+{{r}^{2}}}$.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">State the set of values of $r$ for which the series has a sum to infinity.</div><p style="text-align: right;">[1]</p></li><li><div class="list-item-paragraph">Given that $r=-0.9$, find the least value of $n$ for the sum of $4n$ terms to be greater than $99\%$ of the sum to infinity.</div><p style="text-align: right;">[3]</p></li></ol>

The series $a+ar-a{{r}^{2}}-a{{r}^{3}}+a{{r}^{4}}+a{{r}^{5}}-a{{r}^{6}}-a{{r}^{7}}+$…, where $a&gt;0$, has its $k$th term, ${{T}_{k}}$, defined by ${{T}_{k}}=\left\{ \begin{matrix}
a{{r}^{k-1}}, \\
-a{{r}^{k-1}}, \\
\end{matrix} \right.$ $\begin{matrix}
\mathrm{if}\,k=4p-3\,\,\mathrm{or}\,\,4p-2 \\
\mathrm{if}\,k=4p-1\,\,\mathrm{or}\,\,4p\,\,\,\,\,\,\, \\
\end{matrix}$, for $p\in {{\mathbb{Z}}^{+}}$,

2023 NYJC MYE P1

<p>Starting from the coordinates $\left( 0,8 \right)$, a particle $P$ moves in the positive $x$-direction and then in the negative $y$-direction alternatively (see Fig. 1). Hence, at the end of the ${{\left( 2k \right)}^{\mathrm{th}}}$ move, $P$ has made $k$ horizontal moves and $k$ vertical moves. For example, at the end of ${{5}^{\mathrm{th}}}$ move, $P$ has made $3$ horizontal moves and $2$ vertical moves and at the end of the ${{6}^{\mathrm{th}}}$ move, $P$ has made $3$ horizontal moves and $3$ vertical moves.</p>
<figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/converted/1760155323369-97tkoy8s38h.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NjAxNTUzMjMzNjktOTd0a295OHMzOGgucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mjc2MzYxMywiZXhwIjoxNzkyNzYzOTEzfQ.u3gek1JhOSl1VHAtsfpfKAFrvfgVVvVZzr_QT0c7qDw" width="400" class="aligncenter" /></figure>
<p>The distance in the first horizontal move is $1$ unit, second horizontal move is $1.8$ units, and each successive horizontal move is $80\%$ more than the previous horizontal move.</p>
<p>The distance in the first vertical move is $2$ units, second vertical move is $5$ units, and each successive vertical move is increased by $3$ units.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Find the exact coordinates of $P$ at the end of the ${{9}^{\mathrm{th}}}$ move, showing your working clearly.</div><p style="text-align: right;">[4]</p></li></ol>
<p>$P$ moves from $A$ to $B$ in the ${{\left( 2k-1 \right)}^{\mathrm{th}}}$ and ${{\left( 2k \right)}^{\mathrm{th}}}$ moves. (see Fig. 2)</p>
<figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/converted/1760155380855-zp6n42o969.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NjAxNTUzODA4NTUtenA2bjQybzk2OS5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkyNzYzNjEzLCJleHAiOjE3OTI3NjM5MTN9.nnxBf4B-I38FLL1xJFmcGnr-jythZDnyIcpnL6M6ayE" width="400" class="aligncenter" /></figure>
<ol class="i-list" start="2"><li><div class="list-item-paragraph">Find, in terms of $k$, the distance travelled by $P$ in the ${{\left( 2k-1 \right)}^{\mathrm{th}}}$ and ${{\left( 2k \right)}^{\mathrm{th}}}$ moves respectively.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Hence, find the least value of $k$ such that the gradient of $AB$ is at least $-0.003$.</div><p style="text-align: right;">[3]</p></li></ol>

Starting from the coordinates $\left( 0,8 \right)$, a particle $P$ moves in the positive $x$-direction and then in the negative $y$-direction alternatively (see Fig. 1). Hence, at the end of the ${{\left( 2k \right)}^{\mathrm{th}}}$ move, $P$ has made $k$ horizontal moves and $k$ vertical moves. For example, at the end of ${{5}^{\mathrm{th}}}$ move, $P$ has made $3$ horizontal moves and $2$ vertical moves and at the end of the ${{6}^{\mathrm{th}}}$ move, $P$ has made $3$ horizontal moves and $3$ vertical moves.

The distance in the first horizontal move is $1$ unit, second horizontal move is $1.8$ units, and each successive horizontal move is $80\%$ more than the previous horizontal move.

The distance in the first vertical move is $2$ units, second vertical move is $5$ units, and each successive vertical move is increased by $3$ units.

$P$ moves from $A$ to $B$ in the ${{\left( 2k-1 \right)}^{\mathrm{th}}}$ and ${{\left( 2k \right)}^{\mathrm{th}}}$ moves. (see Fig. 2)

2022 SAJC Promo Q10

<ol class="abc-list"><li><div class="list-item-paragraph">Given that the first, third and fourth terms of an arithmetic progression are three consecutive terms of a geometric progression, and that the sum of the first twenty even-numbered terms of the arithmetic progression is $960$, find the common difference of the arithmetic progression.</div><p style="text-align: right;">[4]</p></li><li><div class="list-item-paragraph">Annie puts $\$\,x$ on 1 January 2010 into a bank account which pays compound interest at a rate of $3%$ per month on the last day of each month. On the first day of each subsequent month, she puts in an amount which is $\$\,x$ more than the amount she puts in the previous month. For example, she puts in $\$\,x$ on 1 January, $\$\,2x$ on 1 February and so on.</div><ol class="i-list"><li><div class="list-item-paragraph">Show that the amount of money in the bank account on the last day of March is $\$\frac{103x}{9}\left[103{{\left(1.03\right)}^{3}}-112\right]$.</div><p style="text-align: right;">[4]</p></li><li><div class="list-item-paragraph">Find the least integer value of $x$ so that the amount of interest earned for the first three months of the year 2010 exceeds $\$100$.</div><p style="text-align: right;">[2]</p></li></ol></li></ol>

Annuities and Amortization

2010 RVHS P1 Q10

<p>Find how many positive integers, less than $1000$, are</p>
<ol class="i-list"><li data-list-item-id="item-1760090900996-lun2fpris" data-list-item-path="item-1760090900996-lun2fpris"><div class="list-item-paragraph">odd numbers,<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1760090901626-zprnokl2b" data-list-item-path="item-1760090901626-zprnokl2b"><div class="list-item-paragraph"><p>odd numbers which are <strong>not</strong> divisible by $5$.</p><span class="inline-marks">[3]</span></div></li></ol>
<p>Find the sum of the odd numbers, less than $1000$, which are <strong>not</strong> divisible by $5$.<span class="inline-marks">[4]</span></p>

Find how many positive integers, less than $1000$, are

Find the sum of the odd numbers, less than $1000$, which are **not** divisible by $5$.

N2004 P1 Q9

<ol class="abc-list"><li data-list-item-id="item-1760082757072-k5wgz1p51" data-list-item-path="item-1760082757072-k5wgz1p51"><div class="list-item-paragraph"><p>In a convergent geometric progression, the sum of the first $n$ terms is equal to the sum of the remaining terms. Given also that the $(n+1)$ th term is $\frac{1}{2}$, determine the value of the $(2n+1)$th term.</p><span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-1760082757574-zd7jny5m0" data-list-item-path="item-1760082757574-zd7jny5m0"><div class="list-item-paragraph">An arithmetic progression has first term $a$ and common difference $d$. The eighth, third and second term of the progression are successive terms of an infinite geometric progression.</div><div class="list-item-paragraph">If the first term of the geometric progression is $10$, find the sum of the even-numbered terms of the progression.<span class="inline-marks">[4]</span></div><div class="list-item-paragraph"><p>A sequence is formed in which the $n$ th term is given by $\ln \left| {{u}_{n}}{{u}_{n+1}} \right|$, where&nbsp;${{u}_{n}}$ is the $n$ th of the geometric progression.</p></div><div class="list-item-paragraph">Show that the sequence forms an arithmetic progression and state the value of the common difference.<span class="inline-marks">[3]</span></div></li></ol>

2009 NYJC P1 Q9

<p>Eden is a new engineer for the Small Rail Transit (SRT) train. His current project is to devise a new schedule for the train to increase its efficiency. In the current schedule, the train will run for $5$ days in one cycle. For the first day of the first cycle, the train runs for $100$ km. On the ${{2}^{\mathrm{nd}}}$ and on the ${{3}^{\mathrm{rd}}}$ day of each cycle, the train will run $30$ km more than the first day of that cycle. On the ${{4}^{\mathrm{th}}}$ and on the ${{5}^{\mathrm{th}}}$ day of each cycle, the train will run $70$ km more compared to the first day of that cycle. On the first day of the ${{k}^{\mathrm{th}}}$ cycle, the train will run 30 km more compared to the last day of the ${{\left( k-1 \right)}^{\mathrm{th}}}$ cycle.</p>
<ol class="i-list"><li><div class="list-item-paragraph">It was found out that the train broke down after travelling $10\mathrm{ }000$ km. By forming an inequality, state the cycle and the day of the cycle that the train was in during the breakdown.</div><p style="text-align: right;">[6]</p></li></ol>
<p>Eden decides to modify the schedule such that the train will now operate every week from Monday to Saturday. On the first day of operations, a Monday, the train runs for $100$ km. On every Tuesday, Thursday and Saturday, the distance travelled will be increased by $20\%$ compared to the previous operating day. However, on every Monday, Wednesday and Friday, there will be a $8\%$ decrease in the distance travelled compared to the previous operating day.</p>
<ol class="i-list" start="2"><li><div class="list-item-paragraph">Find the total distance travelled by the train in $2n$ days of operations.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Find the minimum number of days that the train has run, such that Eden would have improved the previous schedule (i.e. the total distance run before breakdown is more than $10\mathrm{ }000$ km).</div><p style="text-align: right;">[2]</p></li></ol>

Eden is a new engineer for the Small Rail Transit (SRT) train. His current project is to devise a new schedule for the train to increase its efficiency. In the current schedule, the train will run for $5$ days in one cycle. For the first day of the first cycle, the train runs for $100$ km. On the ${{2}^{\mathrm{nd}}}$ and on the ${{3}^{\mathrm{rd}}}$ day of each cycle, the train will run $30$ km more than the first day of that cycle. On the ${{4}^{\mathrm{th}}}$ and on the ${{5}^{\mathrm{th}}}$ day of each cycle, the train will run $70$ km more compared to the first day of that cycle. On the first day of the ${{k}^{\mathrm{th}}}$ cycle, the train will run 30 km more compared to the last day of the ${{\left( k-1 \right)}^{\mathrm{th}}}$ cycle.

Eden decides to modify the schedule such that the train will now operate every week from Monday to Saturday. On the first day of operations, a Monday, the train runs for $100$ km. On every Tuesday, Thursday and Saturday, the distance travelled will be increased by $20\%$ compared to the previous operating day. However, on every Monday, Wednesday and Friday, there will be a $8\%$ decrease in the distance travelled compared to the previous operating day.

2018 YJC P1 Q8

<p>A convergent geometric sequence of positive terms, $G$ has a non-zero first term $a$ and common ratio $r$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">The sum of the first $n$ odd-numbered terms of $G$ is equal to the sum of all terms after the $\left( 2n-1 \right)$th term of $G$. Show that $2{{r}^{2n}}+{{r}^{2n-1}}-1=0$.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">In another sequence $H$, each term is the reciprocal of the corresponding term of $G$. If the $n$th term of $G$ and $H$ is denoted by ${{u}_{n}}$ and ${{v}_{n}}$ respectively, show that a new sequence whose $n$th term is $\ln \left( \frac{{{u}_{n}}}{{{v}_{n}}} \right)$, is an arithmetic progression.</div><p style="text-align: right;">[2]</p></li></ol>

A convergent geometric sequence of positive terms, $G$ has a non-zero first term $a$ and common ratio $r$.

2018 SRJC P1 Q6

<p>A geometric progression has first term $a$ and common ratio $r$, and an arithmetic progression has first term $b$ and common difference $d$, where $a$,$b$,$d$ and $r$ are non-zero real numbers. The first, third and eighth term of the geometric progression are equal to the second, third and fifth term of the arithmetic progression respectively.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Show that ${{r}^{7}}-3{{r}^{2}}+2=0$.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Find the values of $r$, giving your answer correct to $5$ decimal places. Hence explain why the sum to infinity of the geometric progression exists.</div><p style="text-align: right;">[2]</p></li></ol>
<p>It is now given that $a=12$ and $r$ is the positive value found in part (ii).</p>
<ol class="i-list" start="3"><li><div class="list-item-paragraph">Let $E$ be the sum of the first $n$ even-numbered terms of the arithmetic progression and ${{S}_{\infty }}$ be the sum to infinity of the geometric progression. Find the largest value of $n$ such that the difference between $E$ and ${{S}_{\infty }}$ is less than $1000$.</div><p style="text-align: right;">[6]</p></li></ol>

A geometric progression has first term $a$ and common ratio $r$, and an arithmetic progression has first term $b$ and common difference $d$, where $a$,$b$,$d$ and $r$ are non-zero real numbers. The first, third and eighth term of the geometric progression are equal to the second, third and fifth term of the arithmetic progression respectively.

It is now given that $a=12$ and $r$ is the positive value found in part (ii).

2021 TMJC 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 $2{{n}^{2}}+4n$.

Reverse Iteration of Recurrence Relations

2022 VJC J1 MYE Q5

<p>A finite sequence $\{{{a}_{n}}\}$ has $50$ terms and is such that ${{a}_{n+1}}={{a}_{n}}+0.15$ for $n=1,\,\,2,\,\,3,.....\,\,49$</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Given that ${{a}_{50}}=99{{a}_{1}}$ show that ${{a}_{1}}=0.075$.</span></div><p style="text-align: right;">[2]</p></li><li><div class="list-item-content"><span class="list-item-text">Find, without using a calculator, the value of $\sum\limits_{n=1}^{50}{{{a}_{n}}}$.</span></div><p style="text-align: right;">[2]</p></li></ol>
<p>Another infinite sequence $\{{{b}_{m}}\}$ is such that ${{b}_{1}}={{a}_{50}}$ and $\frac{{{b}_{m}}}{{{b}_{m-1}}}=0.98$ for $m\ge 2$.</p>
<ol class="i-list" start="3"><li><div class="list-item-content"><span class="list-item-text">Determine the smallest value of $k$ such that ${{b}_{k}}<{{a}_{25}}$.</span></div><p style="text-align: right;">[2]</p></li><li><div class="list-item-content"><span class="list-item-text">Find the least value of $h$ such that the sum of the first $h$ terms of $\{{{b}_{m}}\}$ is more than $99%$ of its sum to infinity.</span></div><p style="text-align: right;">[2]</p></li><li><div class="list-item-content"><span class="list-item-text">If $\frac{{{b}_{m}}}{{{b}_{m-1}}}=-0.98$ instead, find $\sum\limits_{m=0}^{\infty }{{{b}_{1+3m}}}$.</span></div><p style="text-align: right;">[2]</p></li></ol>

A finite sequence $\{{{a}_{n}}\}$ has $50$ terms and is such that ${{a}_{n+1}}={{a}_{n}}+0.15$ for $n=1,\,\,2,\,\,3,.....\,\,49$

Given that ${{a}_{50}}=99{{a}_{1}}$ show that ${{a}_{1}}=0.075$.

Find, without using a calculator, the value of $\sum\limits_{n=1}^{50}{{{a}_{n}}}$.

Another infinite sequence $\{{{b}_{m}}\}$ is such that ${{b}_{1}}={{a}_{50}}$ and $\frac{{{b}_{m}}}{{{b}_{m-1}}}=0.98$ for $m\ge 2$.

Determine the smallest value of $k$ such that ${{b}_{k}}<{{a}_{25}}$.

Find the least value of $h$ such that the sum of the first $h$ terms of $\{{{b}_{m}}\}$ is more than $99%$ of its sum to infinity.

If $\frac{{{b}_{m}}}{{{b}_{m-1}}}=-0.98$ instead, find $\sum\limits_{m=0}^{\infty }{{{b}_{1+3m}}}$.

2012 DHS P2 Q4

<p>A geometric series is given by</p>
<p style="text-align: center">$1+\frac{{{x}^{2}}-5}{4}+\frac{{{\left( {{x}^{2}}-5 \right)}^{2}}}{16}+...+\frac{{{\left( {{x}^{2}}-5 \right)}^{n}}}{{{4}^{n}}}+...$</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Find the range of values of $x$ such that the series converges.</span></div><p style="text-align: right;">[2]</p></li><li><div class="list-item-content"><span class="list-item-text">If the ratio of the sum of the first five even-numbered terms to the sum of the first five odd-numbered terms is $\frac{x}{3}$, determine the sum to infinity of the geometric series.</span></div><p style="text-align: right;">[3]</p></li></ol>

$1+\frac{{{x}^{2}}-5}{4}+\frac{{{\left( {{x}^{2}}-5 \right)}^{2}}}{16}+...+\frac{{{\left( {{x}^{2}}-5 \right)}^{n}}}{{{4}^{n}}}+...$

Find the range of values of $x$ such that the series converges.

If the ratio of the sum of the first five even-numbered terms to the sum of the first five odd-numbered terms is $\frac{x}{3}$, determine the sum to infinity of the geometric series.

2015 CJC Promo Q4 (c)

<p>A geometric series, $G$, has common ratio $r$, $r\ne 1$, and an arithmetic series, $A$, has a non-zero first term $a$.</p>
<p>The first three terms of $G$ are equal to the seventh, third and first term of $A$ respectively.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Show that $2{{r}^{2}}-3r+1=0$.</span></div><p style="text-align: right;">[2]</p></li><li><div class="list-item-content"><span class="list-item-text">Deduce that $G$ is convergent.</span></div><p style="text-align: right;">[2]</p></li><li><div class="list-item-content"><span class="list-item-text">Find the sum to infinity of the even-numbered terms of $G$ in terms of $a$.</span></div><p style="text-align: right;">[2]</p></li></ol>

A geometric series, $G$, has common ratio $r$, $r\ne 1$, and an arithmetic series, $A$, has a non-zero first term $a$.

The first three terms of $G$ are equal to the seventh, third and first term of $A$ respectively.

Find the sum to infinity of the even-numbered terms of $G$ in terms of $a$.

ASRJC APGP Tutorial Q2

<ol class="abc-list"><li data-list-item-id="item-1759051015469-o8k04bgf9" data-list-item-path="item-1759051015469-o8k04bgf9"><div class="list-item-paragraph">The first $3$ terms of a geometric progression are $a$, $b$, $2$ and the first $3$ terms of an arithmetic progression are $2$, $a$, $b$, with non-zero common difference. Find the values of $a$ and $b$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-1759051017134-ujxqbwl61" data-list-item-path="item-1759051017134-ujxqbwl61"><div class="list-item-paragraph">${{u}_{1}}$, ${{u}_{2}}$, ${{u}_{3}}$, ${{u}_{4}}$, …, ${{u}_{2n}}$ is a sequence of $2n$ positive terms, with $n&gt;1$. The odd-numbered terms form an arithmetic progression with common difference $p$ and the even-numbered terms form a geometric progression with common ratio $\frac{6}{5}$. Given that&nbsp;${{u}_{1}}={{u}_{2}}=p$ and the sum of the odd-numbered terms is less than the sum of the even-numbered terms, find the least value of $n$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-1759051018537-soynut0o3" data-list-item-path="item-1759051018537-soynut0o3"><div class="list-item-paragraph">A geometric progression of $n$ terms has first term $q$ and common ratio $r$, where&nbsp;$q$ is non-zero and $r\ne 1$. For&nbsp;$k\le n$, find the difference between the sum of the last $k$ terms and the sum of the first $k$ terms, simplifying your answer.<span class="inline-marks">[5]</span></div></li></ol>

The first $3$ terms of a geometric progression are $a$, $b$, $2$ and the first $3$ terms of an arithmetic progression are $2$, $a$, $b$, with non-zero common difference. Find the values of $a$ and $b$.

${{u}_{1}}$, ${{u}_{2}}$, ${{u}_{3}}$, ${{u}_{4}}$, …, ${{u}_{2n}}$ is a sequence of $2n$ positive terms, with $n>1$. The odd-numbered terms form an arithmetic progression with common difference $p$ and the even-numbered terms form a geometric progression with common ratio $\frac{6}{5}$. Given that ${{u}_{1}}={{u}_{2}}=p$ and the sum of the odd-numbered terms is less than the sum of the even-numbered terms, find the least value of $n$.

A geometric progression of $n$ terms has first term $q$ and common ratio $r$, where $q$ is non-zero and $r\ne 1$. For $k\le n$, find the difference between the sum of the last $k$ terms and the sum of the first $k$ terms, simplifying your answer.

2022 RI P2 Q4

<ol class="abc-list"><li><div class="list-item-paragraph">A company predicts a yearly profit of $\$100\;000$ in the year 2013. The company also predicts that the yearly profit will rise each year by $5\%$.</div><ol class="i-list"><li><div class="list-item-paragraph">Show that the predicted profit in 2015 is $\$110\;250$.<span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">Find the first year in which the yearly predicted profit exceeds $\$200\;000$.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Find, in terms of $m$, the sum of the predicted yearly profits from the year 2015 till the year $\left( 2000+m \right)$, where&nbsp;$m&gt;15$.<span class="inline-marks">[2]</span></div></li></ol></li><li><div class="list-item-paragraph">An arithmetic progression has first term $a$ and common difference $d$, where&nbsp;$a$ and $d$ are non- zero. The second, seventh and ninth terms of the arithmetic progression are consecutive terms of a geometric progression. Find, in terms of $d$, the sum of the first 20 odd-numbered terms of the arithmetic progression.<span class="inline-marks">[4]</span></div></li></ol>

An arithmetic progression has first term $a$ and common difference $d$, where $a$ and $d$ are non- zero. The second, seventh and ninth terms of the arithmetic progression are consecutive terms of a geometric progression. Find, in terms of $d$, the sum of the first 20 odd-numbered terms of the arithmetic progression.

Math Question Bank