A Math

A Math (IGCSE)

E Math

E Math (IGCSE)

H1 Math

H2 Math

IB AA HL Math

IB AA SL Math

IB AI HL Math

IB AI SL Math

IP Math

N(A) A Math

N(A) E Math

PSLE

PSLE Foundation Mathematics

<p>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

Finding Explicit Terms of Linear Recurrences

Geometric Progression

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

Summations Involving APGP

2025 NYJC Promo Q8

A sequence u_n is defined by u_1=5 and u_n+1=2u_n+3 for n in Z+. Prove that u_n=2n+2-3.

A sequence $\{u_n\}$ is defined by $u_1=5$ and $u_{n+1}=2u_n+3$ for $n\in\mathbb Z^+$. Prove that $u_n=2^{n+2}-3$.

Proof by Mathematical Induction

Closed Form of a Recurrence

Mathematical Induction

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

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

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

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

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

Generating Terms from Recurrence Relations

N2012 P1 Q3 [Modified]

<ol class="abc-list"><li><div class="list-item-paragraph">An arithmetic sequence has first term $a$ and common difference $0.5$, where $a$ is an integer. Find the smallest value of $a$ such that the sum of the first $36$ terms is at least $500$.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">A sequence ${{u}_{0}},\ {{u}_{1}},\ {{u}_{2}},\ \ldots $ is given by</div><div class="list-item-paragraph" style="text-align: center">${{u}_{0}}=400$ and ${{u}_{n}}=1.01{{u}_{n-1}}-x$ for $n\ge 1$, where $x$ is an integer.</div><ol class="i-list"><li><div class="list-item-paragraph">Show that ${{u}_{n}}={{1.01}^{n}}\left( 400 \right)-100x\left( {{1.01}^{n}}-1 \right)$.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Given that $x=16$ and $y=1.01{{u}_{k}}$, where $0 < y < 16$, find the value of $k$ and $y$.</div><p style="text-align: right;">[4]</p></li></ol></li></ol>

2024 CJC Promo Q8

<p>A factory gets permission to dispose, at the start of every day, $600$ kg of waste into a stream of water.</p>
<p>The running stream removes $40\%$  of any waste present, by the end of the day.</p>
<p>Determine a simplified expression for the amount of waste present in the stream at the end of the ${{n}^{\mathrm{th}}}$ day.</p>

A factory gets permission to dispose, at the start of every day, $600$ kg of waste into a stream of water.

The running stream removes $40\%$  of any waste present, by the end of the day.

Determine a simplified expression for the amount of waste present in the stream at the end of the ${{n}^{\mathrm{th}}}$ day.

APGP Revision Worksheet Q1

<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>A patient is injected with $150$ mg of a drug for treatment of a disease. Every $8$ hours, $15\%$ of the drug passes out of his bloodstream. To compensate the loss, a further $x$ mg dose is given every $8$ hours.</p>
<ol class="i-list"><li data-list-item-id="item-1760248150400-0epltp4ef" data-list-item-path="item-1760248150400-0epltp4ef"><div class="list-item-paragraph">Show that the amount of drug (mg) in the patient’s bloodstream immediately after the nth dose is given by $\left( 150-\frac{20}{3}x \right){{\left( 0.85 \right)}^{n-1}}+\frac{20}{3}x$.<span class="inline-marks">[4]</span></div></li></ol>
<p>The safety limit of the amount of drug in a person’s bloodstream is $165$ mg.</p>
<ol class="i-list" start="2"><li data-list-item-id="item-1760248166593-pabd8waju" data-list-item-path="item-1760248166593-pabd8waju"><div class="list-item-paragraph">In the case where&nbsp;$x=25$, find the maximum number of doses that can be administered to the patient for the amount of drug to not exceed the safety limit.<span class="inline-marks">[2]</span></div></li></ol>
<p>Another patient is initially injected with $165$mg of the drug. The amount of drug present at time $t$ hours in the bloodstream of the patient is $V$mg. The drug is removed naturally from his bloodstream at a rate $kV$mg/hour for some positive constant $k$.</p>
<ol class="i-list" start="3"><li data-list-item-id="item-1760248184363-9pwsv6491" data-list-item-path="item-1760248184363-9pwsv6491"><div class="list-item-paragraph">Write down and solve a differential equation, giving $V$ in terms of $t$ and $k$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-1760248186138-ah07mlf6k" data-list-item-path="item-1760248186138-ah07mlf6k"><div class="list-item-paragraph"><p>The drug is effective for treatment only if there is at least 85 mg in the body. Given that&nbsp;𝑉 =100 after 24 hours, find the time taken for the amount of drug to drop from 165 mg to the minimum effective level, giving your answer to the nearest hour.</p><span class="inline-marks">[2]</span></div></li></ol>

A patient is injected with $150$ mg of a drug for treatment of a disease. Every $8$ hours, $15\%$ of the drug passes out of his bloodstream. To compensate the loss, a further $x$ mg dose is given every $8$ hours.

The safety limit of the amount of drug in a person’s bloodstream is $165$ mg.

Another patient is initially injected with $165$mg of the drug. The amount of drug present at time $t$ hours in the bloodstream of the patient is $V$mg. The drug is removed naturally from his bloodstream at a rate $kV$mg/hour for some positive constant $k$.

Solving Differential Equations by Separation of Variables

Exponential Growth and Decay Models

2022 ASRJC J2 MYCT P1 Q11

Differential Equations

<ol class="abc-list"><li data-list-item-id="item-1760174551718-1o7huipab" data-list-item-path="item-1760174551718-1o7huipab"><div class="list-item-paragraph">At the beginning of January 2010, Mr Wong’s fish farm has&nbsp;$40000$ fish breeding in a large pond. In the period between January and December each year, the number of fish increases by $9\%$. At the end of December each year, $5400$ fish are harvested and sold at the fish markets for&nbsp;$\$6$ per fish. No fish died or are being poached while he is in business.</div><ol class="i-list"><li data-list-item-id="subitem-1760174557298-4zrvqvude" data-list-item-path="item-1760174551718-1o7huipab/subitem-1760174557298-4zrvqvude"><div class="list-item-paragraph">Find the number of fish just after the $n$ th harvest, giving your answer in the form $A-B{{\left( 1.09 \right)}^{n}}$, where&nbsp;$A$ and $B$ are constants to be determined.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="subitem-1760174560126-11k7jhi9s" data-list-item-path="item-1760174551718-1o7huipab/subitem-1760174560126-11k7jhi9s"><div class="list-item-paragraph">Find the earliest month and year in which there will be no more fish in the pond.<span class="inline-marks">[2]</span></div><div class="list-item-paragraph">Mr Wong has been thinking of retiring since January 2010. Every December just after the harvest, a nearby fish farm offers to buy over Mr Wong’s business by paying $\$8$ per fish for his remaining stock.</div></li><li data-list-item-id="subitem-1760174560629-43i2upvxu" data-list-item-path="item-1760174551718-1o7huipab/subitem-1760174560629-43i2upvxu"><div class="list-item-paragraph">If he wishes to retire by selling off his fish farm business after the $n$th harvest, find the value of $n$ where the total gains is the maximum.<span class="inline-marks">[3]</span></div></li></ol></li><li data-list-item-id="item-1760174610931-gmdkckgy6" data-list-item-path="item-1760174610931-gmdkckgy6"><div class="list-item-paragraph">White-tailed deer can be found in forests of Southern Canada. A wildlife biologist is investigating the change of a population of white-tailed deer of size $x$ (thousands) at time $t$ (years). He observed that on average, their birth rate is $4000$ per year and their death rate is proportional to the square of the population present.</div><ol class="i-list"><li data-list-item-id="subitem-1760174621979-pqkqkoj5h" data-list-item-path="item-1760174610931-gmdkckgy6/subitem-1760174621979-pqkqkoj5h"><div class="list-item-paragraph">When the population was $5000$, the rate of increase of the population at that instant was $1750$ per year. Assuming that&nbsp;$x$ and $t$ are continuous variables, form a differential equation relating $x$ and $t$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="subitem-1760174622826-eheovyzd1" data-list-item-path="item-1760174610931-gmdkckgy6/subitem-1760174622826-eheovyzd1"><div class="list-item-paragraph">There were $2000$ deer in the forest initially. Solve this differential equation to obtain $x$ as an exact function of $t$.<span class="inline-marks">[4]</span></div></li></ol></li></ol>

Modelling with Multiple Rates of Change

2020 ASRJC P1 Q11

<p>Sales agent $A$ started work on 1st June 2020. He plans to acquire $2$ clients in his first month of work and thereafter increase his clientele by $3$ every month.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Find the number of clients agent $A$ would acquire in June 2021.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Find the number of months needed for agent $A$ to acquire at least $500$ clients in total.</div><p style="text-align: right;">[3]</p></li></ol>
<p>Sales agent $B$ has a total $500$ clients on 1st June 2020. Like agent $A$, he also acquires $3$ new clients every month. However, due to strong competition from other sales agents, agent $B$ loses $1\%$ of his clientele at the end of each month when the clients choose not to continue with his service from the following month.</p>
<ol class="i-list" start="3"><li><div class="list-item-paragraph">Show that the clientele size of agent $B$ at the end of $n$th month is given by</div><div class="list-item-paragraph" style="text-align: center">$297+203{{\left( 0.99 \right)}^{n}}$</div><p style="text-align: right;">[4]</p></li><li><div class="list-item-paragraph">The clientele size of agent $B$ will eventually stabilise. State the value of this clientele size.</div><p style="text-align: right;">[1]</p></li><li><div class="list-item-paragraph">Agent $B$ wishes to keep a clientele size of at least $350$. Find the maximum percentage loss of clients he can have at the end of each month in order to meet this target.</div><p style="text-align: right;">[3]</p></li></ol>

Sales agent $A$ started work on 1st June 2020. He plans to acquire $2$ clients in his first month of work and thereafter increase his clientele by $3$ every month.

Sales agent $B$ has a total $500$ clients on 1st June 2020. Like agent $A$, he also acquires $3$ new clients every month. However, due to strong competition from other sales agents, agent $B$ loses $1\%$ of his clientele at the end of each month when the clients choose not to continue with his service from the following month.

2020 RVHS P1 Q10

<p>A shrimp farmer cultivates marine shrimps in tanks. There are $4000$ marine shrimps in a tank at the start of January $2022$. At the start of each month, the farmer will harvest $500$ marine shrimps for sale. At the end of each month, the number of marine shrimps in the tank will increase by $20%$ from the number at the start of the month after the shrimps are harvested for sale.</p>
<ol class="abc-list"><li><ol class="i-list"><li><div class="list-item-paragraph">Find the number of marine shrimps in the tank at the end of February $2022$.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Show that the number of marine shrimps in the tank at the end of $n$ months after the start of January $2022$ is $1000{{\left( 1.2 \right)}^{n}}+3000$.<span class="inline-marks">[4]</span></div></li><li><div class="list-item-paragraph">The tank will be too congested for marine shrimps to grow well when there are more than $8000$ marine shrimps in the tank. Find when will the tank first become too congested for the marine shrimps to grow well.<span class="inline-marks">[2]</span></div></li></ol></li><li><div class="list-item-paragraph">Salt water is added to the tank at the start of each month to ensure that the marine shrimps are growing well. In January $2022$, $25$m$^{3}$ of salt water is added to the tank. In each subsequent month, the amount of salt water added to the tank is $r%$ of the amount of salt water added in the previous month.</div><ol class="i-list"><li><div class="list-item-paragraph">Assuming $r=55$, find the amount of salt water added to the tank on September $2022$.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">The capacity of the tank is $60$m$^{3}$. Find the range of values of $r$ such that the theoretical maximum total amount of salt water added to the tank will never exceed the capacity of the tank.<span class="inline-marks">[3]</span></div></li></ol></li></ol>

A shrimp farmer cultivates marine shrimps in tanks. There are $4000$ marine shrimps in a tank at the start of January $2022$. At the start of each month, the farmer will harvest $500$ marine shrimps for sale. At the end of each month, the number of marine shrimps in the tank will increase by $20%$ from the number at the start of the month after the shrimps are harvested for sale.

Sum to infinity

2022 VJC J1 MYE Q10

<ol class="abc-list"><li><div class="list-item-paragraph">The volume of water, in litres, in a storage tank decreases by $10\%$ each day. However, $90$ litres of water is also pumped into the tank at the end of each day. The volume of water in the tank at the end of $n$ days is denoted by ${{x}_{n}}$ and ${{x}_{0}}$ is the initial volume of water in the tank.</div><ol class="i-list"><li><div class="list-item-paragraph">Write down a recurrence relation to represent the above situation.</div><p style="text-align: right;">[1]</p><div class="list-item-paragraph">Given that $a{{r}^{n-1}}+...+a{{r}^{2}}+ar+a=a\left( \frac{1-r{{\,}^{n}}}{1-r} \right)$,</div></li><li><div class="list-item-paragraph">Show that ${{x}_{n}}={{0.9}^{n}}\left( {{x}_{0}}-900 \right)+900$.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Deduce the limit of ${{x}_{n}}$ as $n$ becomes very large.</div><p style="text-align: right;">[1]</p></li></ol></li></ol>

Modelling with Recurrence Relations

2008 MJC P1 Q8

<p>A farmer owns a parcel of land that is partially covered with weed. The area that is covered with weed increases by ${80}$m${^{2}}$ each week. The farmer decides to start weeding.</p>
<p>At the start of the first week of weeding, ${500}$ m${^{2}}$ of the land is covered with weed.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">In option 1, the farmer removes weed from ${10\%}$ of the area covered with weed at the end of each week.</span></div><ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find the area covered with weed at the end of the second week.</span></div><p style="text-align: right;">[2]</p></li><li><div class="list-item-content"><span class="list-item-text">Show that the area covered with weed at the end of the nth week is given by ${\left( {{0.9}^{n}}\left( 500 \right)+k\left( 1-{{0.9}^{n}} \right) \right)}$m${^{2}}$ where ${k}$ is a constant to be determined.</span></div><p style="text-align: right;">[3]</p></li><li><div class="list-item-content"><span class="list-item-text">Find the area covered with weed at the end of a week in the long run.</span></div><p style="text-align: right;">[1]</p></li></ol></li><li><div class="list-item-content"><span class="list-item-text">In option ${2}$, the farmer removes weed from an area of ${50}$ m${^{2}}$ at the end of the first week. He removes weed from an additional area of ${10}$ m${^{2}}$ at the end of each subsequent week. Thus he removes weed from an area of ${60}$ m${^{2}}$ at the end of the second week, and ${70}$ m${^{2}}$ at the end of the third week, and so on.</span></div><ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Show that the change in the area covered with weed in the ${{{n}^{\mathrm{th}}}}$ week is given by $\left( 40-10n \right)$m${^{2}}$.</span></div><p style="text-align: right;">[1]</p></li><li><div class="list-item-content"><span class="list-item-text">Hence, or otherwise, find the area covered with weed at the end of the nth week in terms of $n$.</span></div><p style="text-align: right;">[3]</p></li></ol></li></ol>

A farmer owns a parcel of land that is partially covered with weed. The area that is covered with weed increases by ${80}$m${^{2}}$ each week. The farmer decides to start weeding.

At the start of the first week of weeding, ${500}$ m${^{2}}$ of the land is covered with weed.

In option 1, the farmer removes weed from ${10\%}$ of the area covered with weed at the end of each week.

In option ${2}$, the farmer removes weed from an area of ${50}$ m${^{2}}$ at the end of the first week. He removes weed from an additional area of ${10}$ m${^{2}}$ at the end of each subsequent week. Thus he removes weed from an area of ${60}$ m${^{2}}$ at the end of the second week, and ${70}$ m${^{2}}$ at the end of the third week, and so on.

2015 DHS P2 Q4

<p>In the controlled environment of Laboratory (Lab) $A$ where cells are cultivated, the number of cells increases by $250\%$ from the start of the week to the end of the week. At the end of each week, $1.2$ million cells are transferred to another laboratory for experiments. The number of cells (in millions) in Lab $A$ at the start of the ${{n}^{\mathrm{th}}}$ week is given by ${{a}_{n}}$.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Show that ${{a}_{n}}={{3.5}^{n-1}}{{a}_{1}}-0.48\left( {{3.5}^{n-1}}-1 \right)$.</span></div><p style="text-align: right;">[3]</p></li></ol>
<p>It is given that Lab $A$ begins operating at the start of the first week with $0.5$ million cells.</p>
<ol class="i-list" start="2"><li><div class="list-item-content"><span class="list-item-text">Find the number of cells in Lab $A$ at the end of the third week, before the cells are transferred to another laboratory.</span></div><p style="text-align: right;">[2]</p></li></ol>
<p>A research requires at least $60$ million cells before it can commence. To facilitate this, Lab $A$ and Lab $B$ are to cultivate and combine all the cells in both the labs by the start of a certain week. Lab B begins operating one week after Lab $A$.</p>
<p>Lab $B$ begins operating at the start of that week with $0.5$ million cells and cultivates $5$ million cells by the end of that week. At the end of each subsequent week, Lab $B$ cultivates $5$ million cells more than in the previous week.</p>
<ol class="i-list" start="3"><li><div class="list-item-content"><span class="list-item-text">In which week can the research first commence?</span></div><p style="text-align: right;">[3]</p></li></ol>

In the controlled environment of Laboratory (Lab) $A$ where cells are cultivated, the number of cells increases by $250\%$ from the start of the week to the end of the week. At the end of each week, $1.2$ million cells are transferred to another laboratory for experiments. The number of cells (in millions) in Lab $A$ at the start of the ${{n}^{\mathrm{th}}}$ week is given by ${{a}_{n}}$.

Show that ${{a}_{n}}={{3.5}^{n-1}}{{a}_{1}}-0.48\left( {{3.5}^{n-1}}-1 \right)$.

It is given that Lab $A$ begins operating at the start of the first week with $0.5$ million cells.

Find the number of cells in Lab $A$ at the end of the third week, before the cells are transferred to another laboratory.

A research requires at least $60$ million cells before it can commence. To facilitate this, Lab $A$ and Lab $B$ are to cultivate and combine all the cells in both the labs by the start of a certain week. Lab B begins operating one week after Lab $A$.

Lab $B$ begins operating at the start of that week with $0.5$ million cells and cultivates $5$ million cells by the end of that week. At the end of each subsequent week, Lab $B$ cultivates $5$ million cells more than in the previous week.

In which week can the research first commence?

2020 NJC P2 Q3

<p>The number of bacteria (in millions) in Pond ${A}$ at the start of the ${n}$th week, before any chemical treatment, is given by ${u_n}$. Pond ${A}$ is treated at the start of each week with Chemical ${A}$, which kills $70\%$ of all bacteria instantly. At the end of each week, $6$ million new bacteria is reproduced.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Write down a recurrence relation of the form ${u_{n+1} = au_n + b}$, where ${a}$ and ${b}$ are constants to be determined.</span></div><p style="text-align: right;">[1]</p></li><li><div class="list-item-content"><span class="list-item-text">Show that ${{u}_{n}}={{0.3}^{n-1}}{{u}_{1}}+\frac{60}{7}\left( 1-{{0.3}^{n-1}} \right)$.</span></div><p style="text-align: right;">[2]</p></li></ol>
<p>The number of bacteria (in millions) in Pond ${B}$ at the start of the ${n}$th week, before any chemical treatment, is given by ${v_n}$.</p>
<p>Pond ${B}$ is treated at the start of each week with Chemical ${B}$. It is known that ${v_n}$ follows the recurrence relation</p>
<p style="text-align: center">${{v}_{n+1}}=0.01{{v}_{n}}^{2}+6$</p>
<p>It is given that if the sequence ${{v}_{1}}$, ${{v}_{2}}$, ${{v}_{3}}$, … converges to a limit, it converges to either ${\alpha}$ or ${\beta}$, where ${\alpha < \beta}$.</p>
<ol class="i-list" start="3"><li><div class="list-item-content"><span class="list-item-text">Find ${\alpha}$ and ${\beta}$. Explain whether ${v_n}$ necessarily converges to ${\alpha}$ or ${\beta}$.</span></div><p style="text-align: right;">[3]</p></li><li><div class="list-item-content"><span class="list-item-text">If ${u_1 = v_1 = 30}$, determine which chemical would be more effective in killing the bacteria in the long run.</span></div><p style="text-align: right;">[2]</p></li></ol>
<p>Pond ${C}$ is treated with Chemical ${C}$. To account for the bacteria’s increasing resistance to the chemical, the dosage of Chemical ${C}$ is increased by $5$ ml each week. The first dose is $20$ ml.</p>
<ol class="i-list" start="5"><li><div class="list-item-content"><span class="list-item-text">How many weeks does it take to finish the first $3$ litres of chemical in the treatment of Pond ${C}$?</span></div><p style="text-align: right;">[3]</p></li></ol>

The number of bacteria (in millions) in Pond ${A}$ at the start of the ${n}$th week, before any chemical treatment, is given by ${u_n}$. Pond ${A}$ is treated at the start of each week with Chemical ${A}$, which kills $70\%$ of all bacteria instantly. At the end of each week, $6$ million new bacteria is reproduced.

Write down a recurrence relation of the form ${u_{n+1} = au_n + b}$, where ${a}$ and ${b}$ are constants to be determined.

Show that ${{u}_{n}}={{0.3}^{n-1}}{{u}_{1}}+\frac{60}{7}\left( 1-{{0.3}^{n-1}} \right)$.

The number of bacteria (in millions) in Pond ${B}$ at the start of the ${n}$th week, before any chemical treatment, is given by ${v_n}$.

Pond ${B}$ is treated at the start of each week with Chemical ${B}$. It is known that ${v_n}$ follows the recurrence relation

It is given that if the sequence ${{v}_{1}}$, ${{v}_{2}}$, ${{v}_{3}}$, … converges to a limit, it converges to either ${\alpha}$ or ${\beta}$, where ${\alpha < \beta}$.

Find ${\alpha}$ and ${\beta}$. Explain whether ${v_n}$ necessarily converges to ${\alpha}$ or ${\beta}$.

If ${u_1 = v_1 = 30}$, determine which chemical would be more effective in killing the bacteria in the long run.

Pond ${C}$ is treated with Chemical ${C}$. To account for the bacteria’s increasing resistance to the chemical, the dosage of Chemical ${C}$ is increased by $5$ ml each week. The first dose is $20$ ml.

How many weeks does it take to finish the first $3$ litres of chemical in the treatment of Pond ${C}$?

2016 DHS P2 Q3

<p>A sequence ${{u}_{1}}$, ${{u}_{2}}$ , ${{u}_{3}}$, …. , ${{u}_{n}}$, … is such that ${u_1 = 3}$ and ${{u}_{n+1}}=\frac{3}{4}{{u}_{n}}+4$, for ${ u_{n+1} = \frac{3}{4} u_n + 4, \quad \mathrm{for } n \geq 1}$.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Express ${u_n}$ in the form ${16 + A\left(\frac{3}{4}\right)^{n-1}}$, where ${A}$ is a constant to be determined.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Determine whether the sequence ${{u}_{1}}$, ${{u}_{2}}$ , ${{u}_{3}}$, …. , ${{u}_{n}}$, … converges.</span></div></li></ol>

A sequence ${{u}_{1}}$, ${{u}_{2}}$ , ${{u}_{3}}$, …. , ${{u}_{n}}$, … is such that ${u_1 = 3}$ and ${{u}_{n+1}}=\frac{3}{4}{{u}_{n}}+4$, for ${ u_{n+1} = \frac{3}{4} u_n + 4, \quad \mathrm{for } n \geq 1}$.

Express ${u_n}$ in the form ${16 + A\left(\frac{3}{4}\right)^{n-1}}$, where ${A}$ is a constant to be determined.

Determine whether the sequence ${{u}_{1}}$, ${{u}_{2}}$ , ${{u}_{3}}$, …. , ${{u}_{n}}$, … converges.

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