A Math

A Math (IGCSE)

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H1 Math

H2 Math

IB AA HL Math

IB AA SL Math

IB AI HL Math

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IP Math

PSLE

<p>Brand $X$ and Brand $Y$ batteries were tested to find out which brand has a longer battery life. For the test, a toy car was released from a starting point and was allowed to run continuously on a circular racing track until it stopped on its own. Each lap of the circular racing track is $20$ metres.</p>
<ol class="i-list"><li data-list-item-id="imported-item-block-2-item-0" data-list-item-path="imported-item-block-2-item-0"><div class="list-item-paragraph">When the toy car used Brand $X$ battery, it completed the first lap in $15$ seconds and each subsequent lap took $3$ seconds longer than the previous lap.</div><ol class="abc-list"><li data-list-item-id="imported-item-block-2-item-0-sub-0" data-list-item-path="imported-item-block-2-item-0/imported-item-block-2-item-0-sub-0"><div class="list-item-paragraph">In which lap did the toy car first have a lap timing of more than $312.3$ seconds.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="imported-item-block-2-item-0-sub-1" data-list-item-path="imported-item-block-2-item-0/imported-item-block-2-item-0-sub-1"><div class="list-item-paragraph">Given that the toy car ran for&nbsp;$23$ hours before stopping, find the average speed of the car during the last incomplete lap.<span class="inline-marks">[4]</span></div></li></ol></li><li data-list-item-id="imported-item-block-2-item-1" data-list-item-path="imported-item-block-2-item-1"><div class="list-item-paragraph">When the toy car used Brand $Y$ battery, it completed the first lap in $7.68$ seconds and the time taken for each subsequent lap was $2.5\%$ more than the time for the previous lap. Given that the toy car stopped upon completing the $230\text{th}$ lap, determine which brand of battery has a longer battery life. You must show sufficient working to justify your answer.<span class="inline-marks">[2]</span></div></li></ol>
<p>In another test using Brand $Y$ battery, the toy car was made to stop for&nbsp;$5$ seconds each time it completed one lap of the circular racing track. The time taken for the toy car to complete the first lap was still $7.68$ seconds. However, the time taken for the toy car to complete each subsequent lap was $2.5\%$ more than the total time taken for the previous lap and the $5$ seconds stop time. For example the time taken for the toy car to complete the second lap was $1.025 \times (7.68 + 5)$ seconds.</p>
<ol class="i-list" start="3"><li data-list-item-id="imported-item-block-4-item-0" data-list-item-path="imported-item-block-4-item-0"><div class="list-item-paragraph">Show that the time taken for the toy car to complete the $n\text{th}$ lap is $(212.68)(1.025)^{n-1} - 205$ seconds. Hence find the number of laps whereby the time taken to complete a lap is less than one hour.<span class="inline-marks">[4]</span></div></li></ol>

Brand $X$ and Brand $Y$ batteries were tested to find out which brand has a longer battery life. For the test, a toy car was released from a starting point and was allowed to run continuously on a circular racing track until it stopped on its own. Each lap of the circular racing track is $20$ metres.

When the toy car used Brand $X$ battery, it completed the first lap in $15$ seconds and each subsequent lap took $3$ seconds longer than the previous lap.

When the toy car used Brand $Y$ battery, it completed the first lap in $7.68$ seconds and the time taken for each subsequent lap was $2.5\%$ more than the time for the previous lap. Given that the toy car stopped upon completing the $230\text{th}$ lap, determine which brand of battery has a longer battery life. You must show sufficient working to justify your answer.

In another test using Brand $Y$ battery, the toy car was made to stop for $5$ seconds each time it completed one lap of the circular racing track. The time taken for the toy car to complete the first lap was still $7.68$ seconds. However, the time taken for the toy car to complete each subsequent lap was $2.5\%$ more than the total time taken for the previous lap and the $5$ seconds stop time. For example the time taken for the toy car to complete the second lap was $1.025 \times (7.68 + 5)$ seconds.

Show that the time taken for the toy car to complete the $n\text{th}$ lap is $(212.68)(1.025)^{n-1} - 205$ seconds. Hence find the number of laps whereby the time taken to complete a lap is less than one hour.

Problems involving inequalities

Geometric Progression

Financial Arithmetic or Iterative Pattern Recognition

Arithmetic Progression

2026 ACJC17 C1BT1 Q8

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.

Limit of a Sequence and Behaviour of a Sequence

2025 NYJC Promo Q8

<p>An arithmetic series has positive first term $a$ and common difference $d$. The first, fifth and second terms of the arithmetic series are the first three consecutive terms of a geometric series respectively. It is also given that the terms of the arithmetic series are decreasing.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Show that&nbsp;$d = -\frac{7}{16}a$.</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Give a reason why the geometric series converges.</span><span class="inline-marks">[2]</span></div></li></ol>
<p>It is given that the sum to infinity of the geometric series is $\frac{32}{7}$.</p>
<ol class="abc-list" start="3"><li><div class="list-item-content"><span class="list-item-text">Find the sum of the first twenty terms of the arithmetic series.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Given that the sum of all the terms after the $n$th term of the geometric series is greater than $2$% of the sum of the first $n$ terms of the geometric series, find the possible values of $n$.</span><span class="inline-marks">[4]</span></div></li></ol>

An arithmetic series has positive first term $a$ and common difference $d$. The first, fifth and second terms of the arithmetic series are the first three consecutive terms of a geometric series respectively. It is also given that the terms of the arithmetic series are decreasing.

Give a reason why the geometric series converges.

It is given that the sum to infinity of the geometric series is $\frac{32}{7}$.

Find the sum of the first twenty terms of the arithmetic series.

Given that the sum of all the terms after the $n$th term of the geometric series is greater than 2% of the sum of the first $n$ terms of the geometric series, find the possible values of $n$.

Sum to infinity

Inter-sequence Relationships

2025 RI Promo Q11

<ol class="abc-list"><li><div class="list-item-paragraph">The first term of a convergent geometric progression is $3$. The sum of the first five terms of the progression is twice the sum of the first ten terms. Find the sum to infinity of the progression.</div></li><li><div class="list-item-paragraph">The twelfth term of an arithmetic progression is $52$ and the sum of the first eighteen terms is $756$. If the $n$th term of the progression is more than $2000$, find the least possible value of $n$.</div></li></ol>

<p>The sum to infinity of a certain geometric progression is 12, and the sum of the first three terms is $\dfrac{76}{9}$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Find the common ratio of the progression.</div></li><li><div class="list-item-paragraph">Find the least value of $n$ for which the sum of the first $n$ terms is more than 99% of the sum to infinity.</div></li></ol>

The sum to infinity of a certain geometric progression is 12, and the sum of the first three terms is $\dfrac{76}{9}$.

<ol class="abc-list"><li><div class="list-item-paragraph">The sum, ${{S}_{n}}$, of the first $n$ terms of an arithmetic progression is given by ${{S}_{n}}=pn+q{{n}^{2}}$.</div><div class="list-item-paragraph">Given also that ${{S}_{3}}=6$ and ${{S}_{5}}=11$,</div><ol class="i-list"><li><div class="list-item-paragraph">Find the values of $p$ and $q$.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Deduce, or find otherwise, an expression for the $n$-th term and the value of the common difference.</div><p style="text-align: right;">[3]</p></li></ol></li><li><div class="list-item-paragraph">Find the set of values of $\theta $ lying in the interval $-\frac{1}{2}\pi <\theta <\frac{1}{2}\pi $ such that the sum to infinity of the geometric series $1+\sin \theta +{{\sin }^{2}}\theta +\ldots $ is greater than $2$.</div><p style="text-align: right;">[4]</p></li></ol>

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

N1984 P1 Q1

<p>Dendrologists are specialised scientists who study trees and woody plants. Their work is diverse and can encompass various activities related to the identification, classification, biology, and ecology of trees. A group of dendrologists are studying the growth of $2$ species of trees, codenamed Tree Vee and Tree Jay.</p>
<p>In the 1st year, the height of Tree Vee and Tree Jay are both&nbsp;$H$ cm. In the 2nd year, Tree Vee’s height increases by $s$ cm and subsequently, the increase in height every year is $10\%$ less than the previous year’s increase.</p>
<ol class="abc-list"><li data-list-item-id="item-1760748353838-ob8qt2cy4" data-list-item-path="item-1760748353838-ob8qt2cy4"><div class="list-item-paragraph">Show that the height of Tree Vee in the 4th year is given by $\left( H+2.71s \right)$ cm.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1760748354498-ovilxc21a" data-list-item-path="item-1760748354498-ovilxc21a"><div class="list-item-paragraph">Show that the height of Tree Vee in the ${{n}^{\mathrm{th}}}$ year is given by $\left[ H+10s\left( 1-{{0.9}^{n-1}} \right) \right]$ cm.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1760748355053-ow04nd4vr" data-list-item-path="item-1760748355053-ow04nd4vr"><div class="list-item-paragraph">Hence, write down in terms of $H$ and $s$, the theoretical maximum height (in cm) of Tree Vee.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1760748355998-4ovla1o38" data-list-item-path="item-1760748355998-4ovla1o38"><div class="list-item-paragraph">In the 2nd year, Tree Jay’s height increases by $t$ cm and subsequently, the increase in height every year is $0.5$ cm less than the previous year’s increase. Show that the height of Tree Jay in the ${{10}^{\mathrm{th}}}$ year is given by $\left( H-18+9t \right)$ cm.<span class="inline-marks">[2]</span></div></li></ol>
<p>It is now given that&nbsp;$t=20$.</p>
<ol class="abc-list" start="5"><li data-list-item-id="item-1760748358571-xba0beb6d" data-list-item-path="item-1760748358571-xba0beb6d"><div class="list-item-paragraph">After the ${{10}^{\mathrm{th}}}$ year, Tree Jay’s height increases at a constant rate of $7$ cm per year. Express Tree Jay’s height (in cm) in the ${{n}^{\mathrm{th}}}$ year (where&nbsp;$n\ge 11$) in terms of $H$ and $n$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1760748359973-dlg2kmjoa" data-list-item-path="item-1760748359973-dlg2kmjoa"><div class="list-item-paragraph">It is further given that&nbsp;$s=30$, and the 1st year is the year 2024. Find the years in which the heights of Tree Vee and Tree Jay are within 7 cm of each other, after 2034.<span class="inline-marks">[3]</span></div></li></ol>

Dendrologists are specialised scientists who study trees and woody plants. Their work is diverse and can encompass various activities related to the identification, classification, biology, and ecology of trees. A group of dendrologists are studying the growth of $2$ species of trees, codenamed Tree Vee and Tree Jay.

In the 1st year, the height of Tree Vee and Tree Jay are both $H$ cm. In the 2nd year, Tree Vee’s height increases by $s$ cm and subsequently, the increase in height every year is $10\%$ less than the previous year’s increase.

2024 VJC P1 Q11

<p>At a robotic exhibition, a miniature robotic mouse is programmed to walk $2$ different simulation paths from a starting point $O$ to and from a series of points, ${{P}_{1}},\ {{P}_{2}},\ {{P}_{3}},\ \ldots ,$increasingly far away in a straight line. The robotic mouse starts at $O$ and walks stage $1$ from $O$ to ${{P}_{1}}$ and back to $O$, then stage $2$ from $O$ to ${{P}_{2}}$ and back to $O$, and so on. The miniature robotic mouse can walk a maximum distance of $100$ m before its battery runs out.</p>
<div class="graph-block" 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data-width="535" data-height="72" style="width:535px;height:72px;max-width:100%;display:block;margin:0 auto;"></div>
<ol class="i-list"><li data-list-item-id="item-1760696889012-gd8m0fwf4" data-list-item-path="item-1760696889012-gd8m0fwf4"><div class="list-item-paragraph">In simulation $A$, the distances between adjacent points are all $3$ cm (see Fig. 1).</div><ol class="abc-list"><li data-list-item-id="subitem-1760696908647-scjelddxw" data-list-item-path="item-1760696889012-gd8m0fwf4/subitem-1760696908647-scjelddxw"><div class="list-item-paragraph">Find the distance travelled by the robotic mouse that completes the first $12$ stages of simulation $A$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="subitem-1760696915112-sv2p4zfxj" data-list-item-path="item-1760696889012-gd8m0fwf4/subitem-1760696915112-sv2p4zfxj"><div class="list-item-paragraph">Write down an expression for the distance travelled by the robotic mouse that completes $n$ stages of simulation $A$. Hence find the greatest number of stages that the robotic mouse can complete before its battery runs out.<span class="inline-marks">[4]</span></div></li></ol></li></ol>
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<ol class="i-list" start="2"><li data-list-item-id="item-1760696939149-g86asr7wr" data-list-item-path="item-1760696939149-g86asr7wr"><div class="list-item-paragraph">In simulation B, the distances between the points are such that&nbsp;$O{{P}_{1}}=3$ cm,</div><div class="list-item-paragraph">${{P}_{1}}{{P}_{2}}=3$ cm, ${{P}_{2}}{{P}_{3}}=6$ cm and ${{P}_{n}}{{P}_{n+1}}=2{{P}_{n-1}}{{P}_{n}}$ (see Fig. 2). Write down an expression for the distance travelled by the robotic mouse that completes $N$ stages of simulation $B$. Hence find the distance from $O$, and the direction of travel, of the robotic mouse at the instance when the battery runs out in simulation $B$.<span class="inline-marks">[5]</span></div></li></ol>

At a robotic exhibition, a miniature robotic mouse is programmed to walk $2$ different simulation paths from a starting point $O$ to and from a series of points, ${{P}_{1}},\ {{P}_{2}},\ {{P}_{3}},\ \ldots ,$increasingly far away in a straight line. The robotic mouse starts at $O$ and walks stage $1$ from $O$ to ${{P}_{1}}$ and back to $O$, then stage $2$ from $O$ to ${{P}_{2}}$ and back to $O$, and so on. The miniature robotic mouse can walk a maximum distance of $100$ m before its battery runs out.

2015 IJC P1 Q9

<ol class="abc-list"><li data-list-item-id="item-1760418943489-zzk3vgf5z" data-list-item-path="item-1760418943489-zzk3vgf5z"><div class="list-item-paragraph">The sum of the first $n$ terms of a sequence is given by ${{S}_{n}}=\frac{{{n}^{2}}+5n}{8}$. Show that the sequence follows an arithmetic progression with common difference $d$, where&nbsp;$d$ is to be determined.</div><div class="list-item-paragraph">In a geometric progression, the first term is $100$ and its common ratio is $3d$. Find the smallest value of $k$ such that the sum of the first $k$ terms of the arithmetic progression is greater than the sum of the first $30$ terms of the geometric progression.<span class="inline-marks">[6]</span></div></li><li data-list-item-id="item-1760418944390-w7b93f6oq" data-list-item-path="item-1760418944390-w7b93f6oq"><div class="list-item-paragraph">The first and second terms of a geometric sequence are ${{u}_{1}}=a$ and ${{u}_{2}}={{a}^{2}}-a$. If all the terms of the sequence are positive, find the set of values of $a$ for which $\sum\limits_{r=1}^{\infty }{{{u}_{r}}}$ converges.<span class="inline-marks">[2]</span></div><div class="list-item-paragraph">For this sequence, it is known that the sum of all the terms after the ${{n}^{\mathrm{th}}}$ term is equal to the ${{n}^{\mathrm{th}}}$ term. Find the value of a and hence the value of $\sum\limits_{r=1}^{\infty }{{{u}_{r}}}$.<span class="inline-marks">[3]</span></div></li></ol>

2022 VJC P1 Q8

<ol class="abc-list"><li><div class="list-item-paragraph">James puts $\$500$ into an empty safe on 1 January 2023. On the first day of each subsequent month he puts $\$10$ more than in the previous month, so that he puts $\$510$ on 1 February 2023, $\$520$ on 1 March 2023, and so on. On what date will he first have over $\$10,000$ in total in his safe?<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Nora puts $\$x$ on 1 January 2023 into a bank account which pays a compound interest at a rate of $0.5\%$per month on the last day of each month. She puts a further $\$x$ into the account on the first day of each subsequent month. Given that she does not withdraw any money from her account, find the least integer value of $x$ for which her account will accumulate at least $\$50,000$ on 31 December 2027.<span class="inline-marks">[5]</span></div></li></ol>

2023 TJC P1 Q5

<p>Antonio used a credit card to buy a diamond ring to propose to his girlfriend. On 1st October 2022, he charged $\$7500$ to the credit card. The credit card charges a $2\%$ monthly interest rate that is applied on the last day of every month. From 1st November 2022, Antonio started paying off his credit card debt by paying $\$x$ at the start of each month, where $x$ is a constant.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Find the amount of Antonio’s credit card debt left after his first repayment, giving your answer in terms of $x$.</div><p style="text-align: right;">[1]</p></li><li><div class="list-item-paragraph">Show that the amount of Antonio’s credit card debt after his $n$th repayment is</div><div class="list-item-paragraph" style="text-align: center">$\left( 7500-50x \right){{\left( 1.02 \right)}^{n}}+50x$.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Find the minimum amount that Antonio must consistently pay each monthly repayment for him to fully pay off his credit card debt with his $60$th repayment. Find also the amount of credit card interest he would have paid in this case.</div><p style="text-align: right;">[4]</p></li><li><div class="list-item-paragraph">Suppose that up to and including 1st October 2023, Antonio had faithfully repaid $\$500$ a month to service his credit card debt. As a promotional reward, the credit card company decided to lower Antonio’s monthly interest rate to $1\%$ with immediate effect from October 2023, provided he maintains his $\$500$ monthly repayments.</div><div class="list-item-paragraph">Find the date of Antonio’s last repayment to the credit card company with this new interest rate, and the amount of his last repayment.</div><p style="text-align: right;">[4]</p></li></ol>
<p>Antonio and his fiancée are deciding between two venues for their wedding banquet by comparing the costs of the catering at each venue. At the Gomez Hotel, the catering is priced at $\$1500$ for each table of $10$ guests. At the Grande Hotel, the catering is priced as follows:</p>
<figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/converted/1760348977028-1pg2z6ifwaj.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NjAzNDg5NzcwMjgtMXBnMno2aWZ3YWoucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MDk2NzQ3NywiZXhwIjoxNzkwOTY3Nzc3fQ.GYbZEnsxkr40-kzDn9NMhHRouymI9vadbS6Mg5ZOmHk" width="400" class="aligncenter" /></figure>
<ol class="i-list" start="5"><li><div class="list-item-paragraph">How many tables will the couple need at least, for it to be cheaper to hold their wedding banquet at the Grande Hotel?</div><p style="text-align: right;">[3]</p></li></ol>

Antonio used a credit card to buy a diamond ring to propose to his girlfriend. On 1st October 2022, he charged $\$7500$ to the credit card. The credit card charges a $2\%$ monthly interest rate that is applied on the last day of every month. From 1st November 2022, Antonio started paying off his credit card debt by paying $\$x$ at the start of each month, where $x$ is a constant.

Antonio and his fiancée are deciding between two venues for their wedding banquet by comparing the costs of the catering at each venue. At the Gomez Hotel, the catering is priced at $\$1500$ for each table of $10$ guests. At the Grande Hotel, the catering is priced as follows:

2022 ACJC Promo Q11

<p>An arithmetic series has first term $a$ and common difference $d$, where&nbsp;$a&gt;0$ and $d$ is non-zero. The first, eighth and thirteenth terms of the arithmetic series are equal to the first three terms of a geometric series respectively.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Show that&nbsp;$a=-\frac{49}{2}d$.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Show that the geometric series is convergent and find its sum to infinity, $S$, in terms of $a$.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Given that the sum of the first $n$ terms of the arithmetic series is $25$ times the third term of the geometric series, find the value of $n$.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Find the least value of $k$ such that the sum of the first $k$ terms of the geometric series is more than $98\%$ of $S$.<span class="inline-marks">[3]</span></div></li></ol>

An arithmetic series has first term $a$ and common difference $d$, where $a>0$ and $d$ is non-zero. The first, eighth and thirteenth terms of the arithmetic series are equal to the first three terms of a geometric series respectively.

2022 YIJC Promo Q8

<p>The second, fifth and tenth term of an arithmetic series with non-zero common difference are the first three terms of a geometric series respectively.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Find the common ratio.<span class="inline-marks">[3]</span></div></li></ol>
<p>It is further given that the first term of the arithmetic and geometric series are $7$ and $9$ respectively.</p>
<ol class="i-list" start="2"><li><div class="list-item-paragraph">Find the least value of $n$ for which the $n$ th term of the arithmetic series is smaller than the $n$th term of the geometric series by at least $1000$.<span class="inline-marks">[3]</span></div></li></ol>

The second, fifth and tenth term of an arithmetic series with non-zero common difference are the first three terms of a geometric series respectively.

It is further given that the first term of the arithmetic and geometric series are $7$ and $9$ respectively.

2022 CJC Promo Q2

<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 $p\in {{\mathbb{Z}}^{+}}$,

Subsequences of Progressions

2023 NYJC MYE P1

<p>An arithmetic series has first term $a$ and common difference $d$, where $a$ and $d$ are non-zero. The first three terms of a geometric series are equal to the ninth, fifth and second terms respectively of the arithmetic series.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Prove that the geometric series is convergent and find, in terms of $a$, the sum to infinity.</div><p style="text-align: right;">[5]</p></li><li><div class="list-item-paragraph">Given that $a>0$, find the largest value of $n$ for which the sum of the first $n$ terms of the geometric series is less than four fifths of the sum to infinity.</div><p style="text-align: right;">[3]</p></li></ol>

An arithmetic series has first term $a$ and common difference $d$, where $a$ and $d$ are non-zero. The first three terms of a geometric series are equal to the ninth, fifth and second terms respectively of the arithmetic series.

Proving the progression

2011 IJC P1 Q7

<p>A man met with an accident and went into a coma on 10$^{\mathrm{th}}$ January 2013. As a result, he did not pay the bank the outstanding balance of $\$M$ for his credit card bill when it is due for payment on 27$^{\mathrm{th}}$ January 2013. On the 27$^{\mathrm{th}}$ of each month when the payment for the credit card bill is due, the bank will charge a 2% interest on any outstanding balance what is unpaid. After the 2% interest has been added, the bank will still charge an additional late payment charge of $\$L$ monthly.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Express in terms of $L$ and $M$, his outstanding balance on his credit card on 1$^{\mathrm{st}}$ February 2013.</div><p style="text-align: right;">[1]</p></li><li><div class="list-item-paragraph">If the man still remains in coma exactly $n$ months later on the day he met with an accident, show that the accumulated outstanding balance on the man’s credit card is ${{1.02}^{n}}M+50L\left( {{1.02}^{n}}-1 \right)$.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Given that $M=1000$ and $L=55$. Find the least value of $n$ when the accumulated outstanding balance on his credit card first exceeds $\$2010$.</div><p style="text-align: right;">[2]</p></li></ol>

A man met with an accident and went into a coma on 10$^{\mathrm{th}}$ January 2013. As a result, he did not pay the bank the outstanding balance of $\$M$ for his credit card bill when it is due for payment on 27$^{\mathrm{th}}$ January 2013. On the 27$^{\mathrm{th}}$ of each month when the payment for the credit card bill is due, the bank will charge a 2% interest on any outstanding balance what is unpaid. After the 2% interest has been added, the bank will still charge an additional late payment charge of $\$L$ monthly.

2013 ACJC Promo Q15

<p>Mrs Tan plans to start a business which requires a start-up capital of $\$700,000$. She decided to first save $\$200,000$ by depositing money every month into a savings plan. For the remaining $\$500,000$, she intends to take a loan from a finance company.</p>
<p>She deposited $\$3000$ into the savings plan in the first month and on the first day of each subsequent month, she deposited $\$100$ more than the previous month. Mrs Tan will continue depositing money into the savings plan until the total amount in her savings plan reaches $\$200,000$. It is given that this savings plan pays no interest.</p>
<ol class="i-list"><li data-list-item-id="item-1760261886772-61fdl9r6i" data-list-item-path="item-1760261886772-61fdl9r6i"><div class="list-item-paragraph">Find the month in which Mrs Tan’s monthly deposit will exceed $\$6550$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1760261887531-o7awzqx80" data-list-item-path="item-1760261887531-o7awzqx80"><div class="list-item-paragraph">Find the number of months that it will take for Mrs Tan to save $\$200,000$ and hence find the amount that she would have deposited in the last month.<span class="inline-marks">[4]</span></div></li></ol>
<p>After Mrs Tan has saved $\$200,000$, she took a loan of $\$500,000$ from a finance company. To repay the loan from the finance company, Mrs Tan would have to pay a monthly payment of $\$x$ at the beginning of each month, starting from the first month. An interest of $0.3\%$ per month will be charged on the outstanding loan amount at the end of the month.</p>
<ol class="i-list" start="3"><li data-list-item-id="item-1760261910344-wkmf4r3uo" data-list-item-path="item-1760261910344-wkmf4r3uo"><div class="list-item-paragraph">Show that the outstanding amount at the end of ${{n}^{\mathrm{th}}}$ month, after the interest has been charged, is $A\left( {{1.003}^{n}} \right)-Bx\left( {{1.003}^{n}}-1 \right)$, where&nbsp;$A$ and $B$ are exact constants to be determined.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1760261911987-teh2q0rrs" data-list-item-path="item-1760261911987-teh2q0rrs"><div class="list-item-paragraph">Find the amount of $\$x$, to $2$ decimal places, if Mrs Tan wants to fully repay her loan in $8$ years.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1760261912572-h3okmetny" data-list-item-path="item-1760261912572-h3okmetny"><div class="list-item-paragraph">Using the value of $x$ found in part <strong>(iv)</strong>, calculate the total interest that the finance company will earn from Mrs Tan at the end of $8$ years.<span class="inline-marks">[2]</span></div></li></ol>

Mrs Tan plans to start a business which requires a start-up capital of $\$700,000$. She decided to first save $\$200,000$ by depositing money every month into a savings plan. For the remaining $\$500,000$, she intends to take a loan from a finance company.

She deposited $\$3000$ into the savings plan in the first month and on the first day of each subsequent month, she deposited $\$100$ more than the previous month. Mrs Tan will continue depositing money into the savings plan until the total amount in her savings plan reaches $\$200,000$. It is given that this savings plan pays no interest.

After Mrs Tan has saved $\$200,000$, she took a loan of $\$500,000$ from a finance company. To repay the loan from the finance company, Mrs Tan would have to pay a monthly payment of $\$x$ at the beginning of each month, starting from the first month. An interest of $0.3\%$ per month will be charged on the outstanding loan amount at the end of the month.

2021 ACJC Promo Q12

<p>Mary has a monthly income of $\$4000$. She is considering applying for a car loan of $\$40,000$ for $6$ years which charges an interest rate of $3.00\%$ per annum, compounded monthly. Interest is chargeable immediately when the loan sum is drawn out. The monthly payment, $\$m$, is fixed throughout the loan tenure.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Show that the calculated loan balance at the end of the ${{n}^{\mathrm{th}}}$ loan month, after the monthly repayment is made, is given by</div><div class="list-item-paragraph" style="text-align: center">$40000{{\left( \frac{401}{400} \right)}^{n}}-400m\left[ {{\left( \frac{401}{400} \right)}^{n}}-1 \right]$.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">By legislation, banks can approve a car loan only if the monthly repayment does not exceed $15\%$ of an applicant’s monthly income. Prove that Mary will not be able to apply for the car loan.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">If the interest rate for all car loans by the banks is compounded monthly, find the range of interest rates chargeable which will enable Mary to apply for the car loan successfully. Give your answer in the form of $r\%$ per annum, correct to $1$ decimal place.</div><p style="text-align: right;">[3]</p></li></ol>

Mary has a monthly income of $\$4000$. She is considering applying for a car loan of $\$40,000$ for $6$ years which charges an interest rate of $3.00\%$ per annum, compounded monthly. Interest is chargeable immediately when the loan sum is drawn out. The monthly payment, $\$m$, is fixed throughout the loan tenure.

2013 AJC Promo Q12

<p>Two athletes signed up for the $13$ km Nanyang Marathon. In preparation for this marathon, the athletes planned a $16$-day personalised training programme as follows:</p>
<p>Athlete $A$ will run $0.4$ km on the first day of the training programme and for each subsequent day, the distance he will run is increased by $x$ km.</p>
<p>Athlete $B$ will run $0.6$ km on the first day of the training programme and for each subsequent day, the distance he will run is $10\%$ more than the distance ran for the previous day.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Find the minimum value of $x$ such that Athlete $A$ is able to run at least $13$ km on the last day of the training programme.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Athlete $B$ targets to run a total distance of $26$km in these $16$ days. Can he achieve this target? You must show sufficient workings to justify your answer.<span class="inline-marks">[2]</span></div></li></ol>
<p>After $2$ days of training, Athlete $B$ modified his training programme such that on Day $n$, $3\le n\le 16$, he will run $\left[ \frac{10}{21}{{\left( 1.05 \right)}^{n}}+n \right]$km. There are no further changes to his training programme.</p>
<ol class="i-list" start="3"><li><div class="list-item-paragraph">Show that the total distance Athlete $B$ will run by the end of Day $m$, where&nbsp;$3\le m\le 16$, can be expressed as</div><div class="list-item-paragraph" style="text-align: center">$P\left( {{Q}^{m}} \right)+\frac{\left( m-2 \right)\left( m+3 \right)}{2}+R$,</div><div class="list-item-paragraph">where&nbsp;$P$, $Q$ and $R$ are exact constants to be determined.<span class="inline-marks">[5]</span></div></li><li><div class="list-item-paragraph">Athlete $B$ feels that running more than $13$ km per training session may be too strenuous. Assuming he follows the new training programme but limit his running distance to $13$ km per training session, find the total distance he will run in the $16$ days of training.<span class="inline-marks">[4]</span></div></li></ol>

Two athletes signed up for the $13$ km Nanyang Marathon. In preparation for this marathon, the athletes planned a $16$-day personalised training programme as follows:

Athlete $A$ will run $0.4$ km on the first day of the training programme and for each subsequent day, the distance he will run is increased by $x$ km.

Athlete $B$ will run $0.6$ km on the first day of the training programme and for each subsequent day, the distance he will run is $10\%$ more than the distance ran for the previous day.

After $2$ days of training, Athlete $B$ modified his training programme such that on Day $n$, $3\le n\le 16$, he will run $\left[ \frac{10}{21}{{\left( 1.05 \right)}^{n}}+n \right]$km. There are no further changes to his training programme.

Block Sums

2017 NYJC Promo Q12

<p>A bank offers both an ordinary account and a savings account on 1st January 2020 Perlin puts $\$100$ into an ordinary account. On the first day of each subsequent month she saves $\$5$ more than in the previous month, so that she saves $\$105$ on 1st February 2020, $\$110$ on 1st March 2020, and so on. This account pays no interest.</p>
<ol class="i-list"><li><div class="list-item-paragraph">On what date will the amount in Perlin's account first exceed $\$5000$?</div><p style="text-align: right;">[5]</p></li></ol>
<p>On 1st January 2020, Pauline puts $\$100$ into a savings account, and on the first day of each subsequent month she puts another $\$x$ into the account. The interest rate is $0.5\%$ per month, so that on the last day of each month the amount in the account on that day is increased by $0.5\%$.</p>
<ol class="i-list" start="2"><li><div class="list-item-paragraph">Show that the expression for the amount in Pauline’s account on the last day of the ${{n}^{\mathrm{th}}}$ month (where January 2020 is the 1st month, February 2020 is the 2nd month, and so on) is given by</div><div class="list-item-paragraph" style="text-align: center">${{1.005}^{n}}\left( 200x+100 \right)-201x$.</div><p style="text-align: right;">[3]</p><ol class="abc-list"><li><div class="list-item-paragraph">If $x=150$, find the number of complete months for the total amount in Pauline's account to first exceed $\$5,000$</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Pauline wishes for the amount in her account to be at least $\$5,000$ on 31st October 2021 in order to purchase a first-class flight to Japan for her year-end vacation. What is the smallest integer value of $x$ would achieve this?</div><p style="text-align: right;">[2]</p></li></ol></li></ol>

A bank offers both an ordinary account and a savings account on 1st January 2020 Perlin puts $\$100$ into an ordinary account. On the first day of each subsequent month she saves $\$5$ more than in the previous month, so that she saves $\$105$ on 1st February 2020, $\$110$ on 1st March 2020, and so on. This account pays no interest.

On 1st January 2020, Pauline puts $\$100$ into a savings account, and on the first day of each subsequent month she puts another $\$x$ into the account. The interest rate is $0.5\%$ per month, so that on the last day of each month the amount in the account on that day is increased by $0.5\%$.

Math Question Bank