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<p>The sum, $S_n$, of the first $n$ terms of a sequence $u_1,u_2,u_3,\ldots$ is given by \[S_n=\frac16+\frac1{n+3}-\frac2{n+4}.\]</p>
<ol class="abc-list"><li data-list-item-id="nyjc-q05-item01" data-list-item-path="nyjc-q05-item01"><div class="list-item-paragraph">Give a reason why the series $\displaystyle \sum_{r=1}^{\infty}u_r$ converges and write down the value of the sum to infinity.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="nyjc-q05-item02" data-list-item-path="nyjc-q05-item02"><div class="list-item-paragraph">Show that&nbsp;$\displaystyle u_n=\frac{n}{(n+2)(n+3)(n+4)}$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="nyjc-q05-item03" data-list-item-path="nyjc-q05-item03"><div class="list-item-paragraph">It is given that&nbsp;$\displaystyle 5\sum_{r=k+1}^{\infty}u_r=9\sum_{r=1}^{k}u_r$. Find the value of $k$.<span class="inline-marks">[3]</span></div></li></ol>

The sum, $S_n$, of the first $n$ terms of a sequence $u_1,u_2,u_3,\ldots$ is given by \[S_n=\frac16+\frac1{n+3}-\frac2{n+4}.\]

Give a reason why the series $\displaystyle \sum_{r=1}^{\infty}u_r$ converges and write down the value of the sum to infinity.

Show that $\displaystyle u_n=\frac{n}{(n+2)(n+3)(n+4)}$.

It is given that $\displaystyle 5\sum_{r=k+1}^{\infty}u_r=9\sum_{r=1}^{k}u_r$. Find the value of $k$.

Change of Base

Sigma Notation

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

2026 NYJC P1 Q5

APGP

Recurrence Relations and Summation

<p>The sum of the first $n$ terms, where&nbsp;$n\geq1$, of a geometric series is given by \[S_n=\frac54\left[1-\left(-\frac13\right)^n\right].\]</p>
<ol class="abc-list"><li data-list-item-id="hci-q04-item01" data-list-item-path="hci-q04-item01"><div class="list-item-paragraph">Write down the value of the sum to infinity of the geometric series and find the least value of $n$ such that the sum of the first $n$ terms and the sum to infinity differ by less than $10^{-3}$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="hci-q04-item02" data-list-item-path="hci-q04-item02"><div class="list-item-paragraph">Given that&nbsp;$\displaystyle u_k=k\sum_{n=1}^{k}S_n$, find an expression for&nbsp;$u_k$ in terms of $k$ where&nbsp;$k\in\mathbb{Z}^+$.<span class="inline-marks">[4]</span></div></li></ol>

The sum of the first $n$ terms, where $n\geq1$, of a geometric series is given by \[S_n=\frac54\left[1-\left(-\frac13\right)^n\right].\]

Write down the value of the sum to infinity of the geometric series and find the least value of $n$ such that the sum of the first $n$ terms and the sum to infinity differ by less than $10^{-3}$.

Given that $\displaystyle u_k=k\sum_{n=1}^{k}S_n$, find an expression for $u_k$ in terms of $k$ where $k\in\mathbb{Z}^+$.

Sum to infinity

2026 HCI P1 Q4

<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.

Problems involving inequalities

Limit of a Sequence and Behaviour of a Sequence

Geometric Progression

Financial Arithmetic or Iterative Pattern Recognition

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

Arithmetic Progression

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.

Probability - General

Problems with Infinite Outcomes

Probability of a Sequence

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>The sum of the first 5 terms of an arithmetic progression, where the common difference is non-zero, is 105. It is further given that the first, third and fourth terms of the arithmetic progression are the first 3 terms of a geometric progression. Show that the common difference of the arithmetic progression is $-\frac{21}{2}$.<span class="inline-marks">[3]</span></p>
<p>Deduce that the geometric progression is convergent.<span class="inline-marks">[2]</span></p>
<p>Let&nbsp;$S$ be the sum to infinity of the geometric progression, and $A_n$ be the sum to $n$ terms of the arithmetic progression. Find the least value of $n$ such that the sum of $7S$ and $A_n$ is not positive.<span class="inline-marks">[4]</span></p>

The sum of the first 5 terms of an arithmetic progression, where the common difference is non-zero, is 105. It is further given that the first, third and fourth terms of the arithmetic progression are the first 3 terms of a geometric progression. Show that the common difference of the arithmetic progression is $-\frac{21}{2}$.

Deduce that the geometric progression is convergent.

Let $S$ be the sum to infinity of the geometric progression, and $A_n$ be the sum to $n$ terms of the arithmetic progression. Find the least value of $n$ such that the sum of $7S$ and $A_n$ is not positive.

VJC P1 Q13

<ol class="abc-list"><li data-list-item-id="item-a2" data-list-item-path="item-a2"><div class="list-item-paragraph">The sum, $S_n$, of the first $n$ terms of a sequence $u_1, u_2, u_3, \dots$ is given by $S_n = 75 - \frac{3^{n+1}}{5^{n-2}}$.<br>Show that&nbsp;$u_n = k\left(\frac{3}{5}\right)^{n-1}$, where&nbsp;$k$ is a constant to be determined. Find $S_\infty$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-b2" data-list-item-path="item-b2"><div class="list-item-paragraph">Another sequence of numbers $v_1, v_2, v_3, \dots$ is such that&nbsp;$\sum_{i=1}^{n} v_i = \frac{2n}{2n + 1}$.<br>Find the exact value of $\sum_{i=11}^{\infty} v_i$.<span class="inline-marks">[3]</span></div></li></ol>

2024 VJC Promo Q4

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

2024 MI PRU2 P1 Q12

<p>An arithmetic series has first term $a$ and common difference $d$, where&nbsp;$a$ is non-zero. The $8$th, $4$th and $2$nd term 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 increasing.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Show that&nbsp;$a = d$.<span class="inline-marks">[4]</span></div></li></ol>
<p>It is given that the sum of the first $20$ terms of the arithmetic series is $630$.</p>
<ol class="i-list" start="2"><li><div class="list-item-paragraph">Find the value of $a$.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Hence find the exact sum of the first $10$ terms of the geometric series.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Give a reason why the geometric series converges and find the sum to infinity of this series.<span class="inline-marks">[2]</span></div></li></ol>

An arithmetic series has first term $a$ and common difference $d$, where $a$ is non-zero. The $8$th, $4$th and $2$nd term 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 increasing.

It is given that the sum of the first $20$ terms of the arithmetic series is $630$.

2024 MI PRU1 Q8

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

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

2024 MI PRU1 P1 Q1

<p>An arithmetic series has five terms. The first term is $2$ and the last term is $32$. Find the sum of the series.<span class="inline-marks">[4]</span></p>

An arithmetic series has five terms. The first term is $2$ and the last term is $32$. Find the sum of the series.

<p>Find the sum of the arithmetic series<span class="inline-marks">[4]</span></p>
<p style="text-align: center">$17 +27 +37 + \dots 417$.</p>

<p>Find the sum of the infinite geometric series<span class="inline-marks">[4]</span></p>
<p style="text-align: center">$\frac{2}{3} - \frac{4}{9} + \frac{8}{27} - \frac{16}{81} + \dots$</p>

Find the sum of the infinite geometric series

$\frac{2}{3} - \frac{4}{9} + \frac{8}{27} - \frac{16}{81} + \dots$

<p>In an arithmetic sequence, the first term is $5$ and the fourth term is $40$. Find the second term.<span class="inline-marks">[4]</span></p>

In an arithmetic sequence, the first term is $5$ and the fourth term is $40$. Find the second term.

<p>Mr Chan plans to buy a new car in $2031$, which costs $\$120000$ in $2021$. There is a yearly inflation rate of $0.5\%$ on the price of the car.</p>
<ol class="i-list"><li data-list-item-id="item-i" data-list-item-path="item-i"><div class="list-item-paragraph">Show that the cost of the car in $2031$ with inflation is $\$126136.82$.<span class="inline-marks">[2]</span></div></li></ol>
<p>Mr Chan comes across a savings plan in a local bank that pays $1.5\%$ interest on $31$ December of each year. In order to buy the car on $31$ December $2031$, he intends to save $\$y$ every year starting from $1$ January $2022$ for&nbsp;$10$ years with this bank without any withdrawal. On $1$ January $2022$ Mr Chan puts $\$y$ into the bank, and on the $1$ January of each subsequent year, he puts another $\$y$ into the bank.</p>
<ol class="i-list" start="2"><li data-list-item-id="item-ii" data-list-item-path="item-ii"><div class="list-item-paragraph">Find the least value of $\$y$ correct to the nearest dollar.<span class="inline-marks">[5]</span></div></li></ol>
<p>Mr Chan's daughter, Elly, wants to buy a gift for her brother and works out a savings plan with her father. The plan is for&nbsp;$n$ months and Elly will save $\$5$ at the start of every month in a piggy bank. She saves her first $\$5$ on $1$ May $2021$. To help Elly, Mr Chan adds $\$5$ in the piggy bank on the last day of May. For the subsequent months, he puts in $\$5$ more than the previous month in the piggy bank on the last day of each month, so that he puts in $\$5$ in May, $\$10$ in June, and so on. If the gift costs $\$120$, write down an inequality involving $n$ and solve it. Find the date when Elly will save enough money in her piggy bank to buy the gift.<span class="inline-marks">[5]</span></p>

Mr Chan plans to buy a new car in $2031$, which costs $\$120000$ in $2021$. There is a yearly inflation rate of $0.5\%$ on the price of the car.

Mr Chan comes across a savings plan in a local bank that pays $1.5\%$ interest on $31$ December of each year. In order to buy the car on $31$ December $2031$, he intends to save $\$y$ every year starting from $1$ January $2022$ for $10$ years with this bank without any withdrawal. On $1$ January $2022$ Mr Chan puts $\$y$ into the bank, and on the $1$ January of each subsequent year, he puts another $\$y$ into the bank.

Mr Chan's daughter, Elly, wants to buy a gift for her brother and works out a savings plan with her father. The plan is for $n$ months and Elly will save $\$5$ at the start of every month in a piggy bank. She saves her first $\$5$ on $1$ May $2021$. To help Elly, Mr Chan adds $\$5$ in the piggy bank on the last day of May. For the subsequent months, he puts in $\$5$ more than the previous month in the piggy bank on the last day of each month, so that he puts in $\$5$ in May, $\$10$ in June, and so on. If the gift costs $\$120$, write down an inequality involving $n$ and solve it. Find the date when Elly will save enough money in her piggy bank to buy the gift.

2021 JPJC J2 MYE P1 Q10

<p>In a geometric series, $u_1 = \frac{1}{81}$ and $u_4 = \frac{1}{3}$</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Find the value of $r$.</div></li><li><div class="list-item-paragraph">Find the smallest value of $n$ for which $S_n &gt; 40$.</div></li></ol>

In a geometric series, $u_1 = \frac{1}{81}$ and $u_4 = \frac{1}{3}$

<p>Consider the arithmetic sequence $3, 9 ,15 , \dots, 1353$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Write down the common difference.</div></li><li><div class="list-item-paragraph">Find the number of terms in the sequence.</div></li><li><div class="list-item-paragraph">Find the sum of the sequence.</div></li></ol>

Consider the arithmetic sequence $3, 9 ,15 , \dots, 1353$.

<p>In an arithmetic sequence $u_1 = 7$, $u_{20} = 64$ and $u_n = 3709$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Find the value of the common difference.</div></li><li><div class="list-item-paragraph">Find the value of $n$.</div></li></ol>

In an arithmetic sequence $u_1 = 7$, $u_{20} = 64$ and $u_n = 3709$.

<p>In an arithmetic sequence, $u_1 = 2$ and $u_3 = 8$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Find $d$.</div></li><li><div class="list-item-paragraph">Find $u_{20}$.</div></li><li><div class="list-item-paragraph">Find $S_{20}$.</div></li></ol>

In an arithmetic sequence, $u_1 = 2$ and $u_3 = 8$.

Math Question Bank