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<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

APGP

<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

Recurrence Relations and Summation

<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 $u_3 = 8$.

<p>The annual wage of a certain occupation offered by Company $P$ begins with an initial amount of $\$9500$ and increases by $\$400$ every year for&nbsp;$14$ years till it reaches $\$15100$ and remains constant at $\$15100$ thereafter.</p>
<p>Let&nbsp;$p(n) = a_1 + a_2 + a_3 + ... + a_n$, where&nbsp;$a_i$ denotes the annual wage in the $i^{\mathrm{th}}$ year offered by Company $P$ for&nbsp;$i = 1, 2, 3, \dots, n$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Show that&nbsp;$p(n) = \begin{cases} 200n^2 + 9300n & \text{for}\hspace{0.5em} n \le 15, \\ 15100n + A & \text{for}\hspace{0.5em} n &gt; 15, \end{cases}$ </div><div class="list-item-paragraph">where&nbsp;$A$ is a constant to be determined.<span class="inline-marks">[4]</span></div></li><li><div class="list-item-paragraph">The annual wage of the same occupation offered by Company $Q$ begins with the same initial amount of $\$9500$ but it increases at a fixed rate of $r \, \%$ every year.</div><div class="list-item-paragraph">Sam has been offered a job for the same occupation at both Company $P$ and $Q$. Find the value of $r$, correct to $2$ decimal places, such that the total annual wages earned by Sam if he works for Company $P$ for&nbsp;$15$ years is equivalent to the total annual wages earned if he works for Company $Q$ for&nbsp;$12$ years.<span class="inline-marks">[4]</span></div></li></ol>

The annual wage of a certain occupation offered by Company $P$ begins with an initial amount of $\$9500$ and increases by $\$400$ every year for $14$ years till it reaches $\$15100$ and remains constant at $\$15100$ thereafter.

Let $p(n) = a_1 + a_2 + a_3 + ... + a_n$, where $a_i$ denotes the annual wage in the $i^{\mathrm{th}}$ year offered by Company $P$ for $i = 1, 2, 3, \dots, n$.

2014 RI Promo Q6

<p>The cost of Sonia's new car was $\$P$. She accepted an interest-free loan of $\$P$, which she agreed to repay by monthly instalments. The first instalment was $\$1200$. The instalments were increased by $\$50$ per month so the second and third instalments were $\$1250$ and $\$1300$ respectively.</p>
<p>Given that the loan was repaid in $k$ instalments, and that the final instalment was $\$3250$, find the value of $k$ and $P$.</p>

The cost of Sonia's new car was $\$P$. She accepted an interest-free loan of $\$P$, which she agreed to repay by monthly instalments. The first instalment was $\$1200$. The instalments were increased by $\$50$ per month so the second and third instalments were $\$1250$ and $\$1300$ respectively.

Given that the loan was repaid in $k$ instalments, and that the final instalment was $\$3250$, find the value of $k$ and $P$.

Financial Arithmetic or Iterative Pattern Recognition

2009 SAJC Promo Q13(a)

<p>A sequence $a_1, a_2, a_3, \dots$ is such that&nbsp;$a_n &gt; 0$, for&nbsp;$n \in \mathbb{Z}^+$, $n \geq 1$.</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Find $\sum_{n=1}^{N} \ln \left( \frac{a_{n+1}}{a_n} \right)$, and express your answer as a single logarithm.<span class="inline-marks">[2]</span></div></li></ol>
<p>It is also known that this sequence is defined by $\frac{a_{n+1}}{a_n} = \frac{1}{4}$.</p>
<ol class="abc-list" start="2"><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">Given that&nbsp;$\sum_{n=1}^{N} (a_n - a_{n+1})^2 = \frac{k}{16} \sum_{n=1}^{N} (a_n)^2$, find the value of $k$.<span class="inline-marks">[2]</span></div><ol class="i-list"><li data-list-item-id="item-b-i" data-list-item-path="item-b/item-b-i"><div class="list-item-paragraph">show that&nbsp;$\sum_{n=1}^{N} (a_n - a_{n+1})^2 = \frac{3}{5} a_1^2 \left( 1 - \frac{1}{16^N} \right)$, for&nbsp;$N \in \mathbb{Z}^+$, $N \geq 1$,<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-b-ii" data-list-item-path="item-b/item-b-ii"><div class="list-item-paragraph">find $\sum_{n=2}^{\infty} (a_n - a_{n+1})^2$, leaving your answers in terms of $a_1$.<span class="inline-marks">[3]</span></div></li></ol></li></ol>

A sequence $a_1, a_2, a_3, \dots$ is such that $a_n > 0$, for $n \in \mathbb{Z}^+$, $n \geq 1$.

Find $\sum_{n=1}^{N} \ln \left( \frac{a_{n+1}}{a_n} \right)$, and express your answer as a single logarithm.

It is also known that this sequence is defined by $\frac{a_{n+1}}{a_n} = \frac{1}{4}$.

Given that $\sum_{n=1}^{N} (a_n - a_{n+1})^2 = \frac{k}{16} \sum_{n=1}^{N} (a_n)^2$, find the value of $k$.

2023 ACJC Promo Q10

<ol class="abc-list"><li><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><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><div class="list-item-paragraph">The sum of the first $n$ terms of a series is given by ${{S}_{n}}=\frac{1}{3}n\left( n+12 \right)$. By finding an expression for the ${{n}^{\text{th}}}$ 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>

<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.

Geometric Progression

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>

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

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