A Math

A Math (IGCSE)

E Math

H1 Math

H2 Math

IB AA HL Math

IB AA SL Math

IB AI HL Math

IB AI SL Math

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

<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

N2025 P1 Q2

<p>Kai and Lina play a sequence of rounds. Kai wins the first round with probability $\dfrac{3}{5}$. For every round after the first, the conditional probability that the winner of the preceding round also wins the next round is $\dfrac{2}{3}$. The match ends as soon as a player wins two consecutive rounds.</p>
<ol class="i-list"><li data-list-item-id="gp-q2-i" data-list-item-path="gp-q2-i"><div class="list-item-paragraph">The first four possible winning sequences for Kai are $KK$, $LKK$, $KLKK$ and $LKLKK$, where&nbsp;$K$ denotes a round won by Kai and $L$ denotes a round won by Lina. Find the probability of each sequence.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="gp-q2-ii" data-list-item-path="gp-q2-ii"><div class="list-item-paragraph">Hence, using the sum to infinity of a geometric progression, find the probability that Kai wins the match.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="gp-q2-iii" data-list-item-path="gp-q2-iii"><div class="list-item-paragraph">Given that Lina wins the first round, find the probability that Kai wins the match.<span class="inline-marks">[2]</span></div></li></ol>

Kai and Lina play a sequence of rounds. Kai wins the first round with probability $\dfrac{3}{5}$. For every round after the first, the conditional probability that the winner of the preceding round also wins the next round is $\dfrac{2}{3}$. The match ends as soon as a player wins two consecutive rounds.

Conditional Probability

Problems with Infinite Outcomes

Probability of a Sequence

Sum to infinity

Probability

<p>Ava and Ben take turns throwing a dart at a target. The probability that Ava hits the target on any throw is $\dfrac{2}{5}$, while the probability that Ben hits it on any throw is $\dfrac{1}{3}$. All throws are independent. A player wins as soon as that player hits the target. Ava throws first.</p>
<ol class="i-list"><li data-list-item-id="gp-q1-i" data-list-item-path="gp-q1-i"><div class="list-item-paragraph">Show that the probability that both players miss in one pair of consecutive throws, one by Ava followed by one by Ben, is $\dfrac{2}{5}$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="gp-q1-ii" data-list-item-path="gp-q1-ii"><div class="list-item-paragraph">Find the probability that Ava wins the game.<span class="inline-marks">[3]</span></div></li></ol>
<p>In a second game, a biased coin is used to choose who throws first. Ben is twice as likely as Ava to throw first. The same hit probabilities and rules apply.</p>
<ol class="i-list" start="3"><li data-list-item-id="gp-q1-iii" data-list-item-path="gp-q1-iii"><div class="list-item-paragraph">Find the probability that Ava wins the second game.<span class="inline-marks">[3]</span></div></li></ol>

Ava and Ben take turns throwing a dart at a target. The probability that Ava hits the target on any throw is $\dfrac{2}{5}$, while the probability that Ben hits it on any throw is $\dfrac{1}{3}$. All throws are independent. A player wins as soon as that player hits the target. Ava throws first.

In a second game, a biased coin is used to choose who throws first. Ben is twice as likely as Ava to throw first. The same hit probabilities and rules apply.

<p>Consider the function</p>
<p style="text-align: center">\[S_n(x) = 1 + \frac{2}{1+x} + \left( \frac{2}{1+x} \right)^2 + \dots + \left( \frac{2}{1+x} \right)^n, \quad x \neq -1.\]</p>
<p>Determine the set of values of $x$ for which the sum to infinity $S(x)$ exists.</p>

\[S_n(x) = 1 + \frac{2}{1+x} + \left( \frac{2}{1+x} \right)^2 + \dots + \left( \frac{2}{1+x} \right)^n, \quad x \neq -1.\]

Determine the set of values of $x$ for which the sum to infinity $S(x)$ exists.

Sum to infinity with undetermined range

<p>Evaluate in terms of $n$ the sum:</p>
<p style="text-align: center">\[3 + 33 + 333 + 3333 + \dots + \underbrace{3 \dots 3}_{n \text{ copies}}\]</p>

\[3 + 33 + 333 + 3333 + \dots + \underbrace{3 \dots 3}_{n \text{ copies}}\]

Geometric sum of a number pattern

<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 $n$.

Inter-sequence Relationships

2025 RI Promo Q11

<p>Show that the geometric series $9$, $-3$, $1$, ..... is convergent. Find the sum to infinity.</p>

Show that the geometric series $9$, $-3$, $1$, ..... is convergent. Find the sum to infinity.

<p>Show that the geometric series $5$, $\dfrac{5}{2}$, $\dfrac{5}{4}$, .... is convergent. Find the sum to infinity.</p>

Show that the geometric series $5$, $\dfrac{5}{2}$, $\dfrac{5}{4}$, .... is convergent. Find the sum to infinity.

<p>Find the two possible values of $x$ if $-5$, $x$, $-45$ are three consecutive terms of a geometric progression.</p>

Find the two possible values of $x$ if $-5$, $x$, $-45$ are three consecutive terms of a geometric progression.

<p>Find the two possible values of $x$ if $3$, $x$, $48$ are three consecutive terms of a geometric progression.</p>

Find the two possible values of $x$ if $3$, $x$, $48$ are three consecutive terms of a geometric progression.

<p>Find the common ratio and write down the $n$th term of the geometric progression $8$,$ -4$, $2$, .... . </p>
<p>Find also the sum of the first $n$ terms.</p>

Find the common ratio and write down the $n$th term of the geometric progression $8$,$ -4$, $2$, .... . 

Find also the sum of the first $n$ terms.

<p>Find the common ratio and write down the $n$th term of the geometric progression $3$, $12$, $48$, ..... . </p>
<p>Find also the sum of the first $n$ terms.</p>

Find the common ratio and write down the $n$th term of the geometric progression $3$, $12$, $48$, ..... . 

<p>A geometric series has first term $a$ and common ratio $r$, where $r<0$. The sum to infinity of the series is $18$ and its third term is $\frac{8}{3}$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Show that $27{{r}^{3}}-27{{r}^{2}}+4=0$.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Find the values of $r$ and $a$.</div><p style="text-align: right;">[1]</p></li></ol>
<p>The $n$ th term of the series is ${{u}_{n}}$.</p>
<ol class="abc-list" start="3"><li><div class="list-item-paragraph">It is given that</div><div class="list-item-paragraph" style="text-align: center">$80\sum\limits_{r=k+1}^{\infty }{{{u}_{r}}}=\sum\limits_{r=1}^{k}{{{u}_{r}}}$.</div><div class="list-item-paragraph">Find the value of $k$.</div><p style="text-align: right;">[3]</p></li></ol>

A geometric series has first term $a$ and common ratio $r$, where $r<0$. The sum to infinity of the series is $18$ and its third term is $\frac{8}{3}$.

The $n$ th term of the series is ${{u}_{n}}$.

Manipulating Summation Limits

N2024 P1 Q5

<p>Consider the complex series ${{{e}^{\mathrm{i}\theta }}+\frac{1}{2}{{e}^{\mathrm{2i}\theta }}+\frac{1}{4}{{e}^{\mathrm{3i}\theta }}+\frac{1}{8}{{e}^{\mathrm{4i}\theta }}+…}$</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Find ${{{u}_{r}}}$ in terms of ${r}$, where&nbsp;${{{u}_{r}}}$ is the ${r}$th term of the series.<span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">Show that the series is geometric and convergent. Hence, state the sum to infinity.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">By expressing the sum to infinity in terms of ${\sin \theta }$ and ${\cos \theta }$, show that</div><div class="list-item-paragraph" style="text-align: center">${\cos \theta +\frac{1}{2}\cos 2\theta +\frac{1}{4}\cos 3\theta +…=\frac{4\cos \theta -2}{5-4\cos \theta }}$.<span class="inline-marks">[5]</span></div></li></ol>

Consider the complex series ${{{e}^{\mathrm{i}\theta }}+\frac{1}{2}{{e}^{\mathrm{2i}\theta }}+\frac{1}{4}{{e}^{\mathrm{3i}\theta }}+\frac{1}{8}{{e}^{\mathrm{4i}\theta }}+…}$

Properties of Conjugate Complex Numbers

Properties from Exponential or Trigonometric form

2023 TGM P1 Q8

Complex Numbers

<ol class="abc-list"><li><div class="list-item-paragraph">Show that&nbsp;${\left| {{\mathrm{e}}^{\mathrm{i}\theta }} \right|=1}$.<span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">Consider the geometric series ${1+\frac{1}{3}{{\mathrm{e}}^{\mathrm{i}\theta }}+\frac{1}{9}{{\mathrm{e}}^{2\mathrm{i}\theta }}+…}$</div><div class="list-item-paragraph">Write down the common ratio, ${z}$, of the series and show that&nbsp;${\left| z \right|=\frac{1}{3}}$.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Find an expression for the sum to infinity of this series.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Hence, show that&nbsp;${\sin \theta +\frac{1}{3}\sin 2\theta +\frac{1}{9}\sin 3\theta +…=\frac{9\sin \theta }{10-6\cos \theta }}$.<span class="inline-marks">[8]</span></div></li></ol>

<p>On his ${{45}^{\mathrm{th}}}$ birthday on $1$ July $2024$, Mr Eu purchases an annuity plan called EuRetire for a principal sum of $\$x$. The plan works like this:</p>
<p><strong>Phase 1:</strong> The plan pays compound interest at a rate of $3\%$ per annum on the last day of June each year, until Mr Eu turns $65$. The total sum at the end of Phase 1 is called the “accumulated sum”.</p>
<p><strong>Phase 2:</strong> Upon turning $65$, Mr Eu receives a monthly payout, $\$y$, on the ${{1}^{\mathrm{st}}}$ of each month, from the accumulated sum. The amount remaining in the plan continues to draw interest at $0.2\%$ per month on the last day of each month.</p>
<ol class="abc-list"><li data-list-item-id="item-1760753493990-nobajnq7h" data-list-item-path="item-1760753493990-nobajnq7h"><div class="list-item-paragraph">Write down an expression for the accumulated sum at the end of Phase 1, giving your answer to $5$ significant figures.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1760753494538-0yukkoe6j" data-list-item-path="item-1760753494538-0yukkoe6j"><div class="list-item-paragraph">Show that the amount remaining in the plan at the end of the ${{n}^{\mathrm{th}}}$ month of Phase 2, after interest has been credited, is</div><div class="list-item-paragraph" style="text-align: center">$\$1.8061\left({{1.002}^{n}}\right)x-501\left({{1.002}^{n}}-1\right)y$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-1760753495850-omow3pwib" data-list-item-path="item-1760753495850-omow3pwib"><div class="list-item-paragraph">Mr Eu wishes to receive $\$1000$ a month in Phase 2 for at least $10$ years. What is the minimum principal sum, correct to the nearest dollar, he would need?<span class="inline-marks">[2]</span></div></li></ol>

On his ${{45}^{\mathrm{th}}}$ birthday on $1$ July $2024$, Mr Eu purchases an annuity plan called EuRetire for a principal sum of $\$x$. The plan works like this:

**Phase 1:** The plan pays compound interest at a rate of $3\%$ per annum on the last day of June each year, until Mr Eu turns $65$. The total sum at the end of Phase 1 is called the “accumulated sum”.

**Phase 2:** Upon turning $65$, Mr Eu receives a monthly payout, $\$y$, on the ${{1}^{\mathrm{st}}}$ of each month, from the accumulated sum. The amount remaining in the plan continues to draw interest at $0.2\%$ per month on the last day of each month.

2024 EJC P1 Q11

<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>Customers who open a savings account with SRF bank will earn an interest rate of $0.2\%$ per month based on the amount in the account on the last day of each month. The interest is added to the account at the end of each month.</p>
<ol class="abc-list"><li data-list-item-id="item-1760689764559-ualjpv8ke" data-list-item-path="item-1760689764559-ualjpv8ke"><div class="list-item-paragraph">Peter has a savings account with the SRF bank. He puts $\$2000$ into his savings account at the beginning of each month starting from ${{1}^{\mathrm{st}}}$ January 2023. Find the amount he has in his savings account on ${{31}^{\mathrm{st}}}$ December 2027 after the interest has been added if no withdrawal is made. Show your workings clearly.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-1760689765331-qpxbpom5w" data-list-item-path="item-1760689765331-qpxbpom5w"><div class="list-item-paragraph"><p>Sandy, another customer of SRF bank, has retired from work starting ${{1}^{\mathrm{st}}}$ January $2023$. Since then, she will no longer do any deposit of money into her own savings account with the bank. Instead, she will be using the money saved in that account for her monthly spending. She has&nbsp;$\$750\,000$ in her SRF bank savings account on ${{1}^{\mathrm{st}}}$ January $2023$.</p></div><ol class="i-list"><li data-list-item-id="subitem-1760689790367-d4jrkulvn" data-list-item-path="item-1760689765331-qpxbpom5w/subitem-1760689790367-d4jrkulvn"><div class="list-item-paragraph"><p>As she still has some money with her for her daily spending in January $2023$, she will only commence her monthly withdrawal of $\$k$ from her savings account in the middle of each month, starting from February $2023$ onwards. Show that the amount of money in her savings account, at the end of $n$ complete months from ${{1}^{\mathrm{st}}}$ January $2023$, after the interest has been added, is given by ${{1.002}^{n}}\left( \$750\,\,000\right)-\$501k\left({{1.002}^{n-1}}\right)+\$501k$.</p><span class="inline-marks">[4]</span></div></li><li data-list-item-id="subitem-1760689791260-a0sj3x2o6" data-list-item-path="item-1760689765331-qpxbpom5w/subitem-1760689791260-a0sj3x2o6"><div class="list-item-paragraph"><p>Taking $k$ to be $3000$, find the amount of money left in her savings account immediately after her last complete withdrawal of $\$ 3000$.</p><span class="inline-marks">[4]</span></div></li></ol></li></ol>

Customers who open a savings account with SRF bank will earn an interest rate of $0.2\%$ per month based on the amount in the account on the last day of each month. The interest is added to the account at the end of each month.

Math Question Bank