A Math

A Math (IGCSE)

E Math

H1 Math

H2 Math

IB HL Math

IB SL Math

IP Math

<p>The $n \mathrm{th}$ term of a sequence is given by $ T_{n} = \frac{3n +2}{47 - 2n}$</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Find $T_{3}$.</div></li><li><div class="list-item-paragraph">Given that&nbsp;$ T_{k} = 13$, find the value of $k$.</div></li><li><ol class="i-list"><li><div class="list-item-paragraph">Explain why $3n + 2$ is always positive.</div></li><li><div class="list-item-paragraph">Hence or otherwise, find the least value of $n$ for which $T_{n} &lt; 0$.</div></li></ol></li></ol>

The $n \mathrm{th}$ term of a sequence is given by $ T_{n} = \frac{3n +2}{47 - 2n}$

General Term of a Sequence

2022 TKSS EOY P2 Q3

Number Patterns

<p>The dots in each of the following diagrams form the border of an equilateral triangle which surrounds the inner cross(es) that is/are required to form the triangular pattern. The number of layer of dots of each triangle can be obtained by counting the dots on the side of the triangle.</p>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1770638928788-gebcs0w4r97.png?token=eyJraWQiOiJzdG9yYWdlLXVybC1zaWduaW5nLWtleV9iZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NzA2Mzg5Mjg3ODgtZ2ViY3MwdzRyOTcucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4NTcxMDU5MCwiZXhwIjoxNzg1NzEwODkwfQ.8u_QD4wbeBOFZPR7hRtVoVe8RD3bjmEvyouGDC3kYvw" style="max-width: min(500px, 100%); height: auto; display: block;" alt="Diagrams of triangular patterns with border dots and inner crosses for layers $3$, $4$, and $5$." class="aligncenter custom-width" /></figure></div>
<p>The table below gives information about the numbers of dots and crosses in triangles of various layers.</p>
<div class="content-block content-block--table"><div class="table-container"><table style="width: 100%;"><thead><tr><th style="text-align: center; --cell-justify: center">No. of layers</th><th style="text-align: center; --cell-justify: center">$3$</th><th style="text-align: center; --cell-justify: center">$4$</th><th style="text-align: center; --cell-justify: center">$5$</th><th style="text-align: center; --cell-justify: center">$6$</th><th style="text-align: center; --cell-justify: center">$\dots$</th><th style="text-align: center; --cell-justify: center">$u$</th><th style="text-align: center; --cell-justify: center">$\dots$</th><th style="text-align: center; --cell-justify: center">$28$</th><th style="text-align: center; --cell-justify: center">$\dots$</th><th style="text-align: center; --cell-justify: center">$34$</th><th style="text-align: center; --cell-justify: center">$\dots$</th><th style="text-align: center; --cell-justify: center">$n$</th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center">No. of border dots</td><td style="text-align: center; --cell-justify: center">$6$</td><td style="text-align: center; --cell-justify: center">$9$</td><td style="text-align: center; --cell-justify: center">$12$</td><td style="text-align: center; --cell-justify: center">$15$</td><td style="text-align: center; --cell-justify: center">$\dots$</td><td style="text-align: center; --cell-justify: center">$33$</td><td style="text-align: center; --cell-justify: center">$\dots$</td><td style="text-align: center; --cell-justify: center">$v$</td><td style="text-align: center; --cell-justify: center">$\dots$</td><td style="text-align: center; --cell-justify: center">$99$</td><td style="text-align: center; --cell-justify: center">$\dots$</td><td style="text-align: center; --cell-justify: center">$y$</td></tr><tr><td style="text-align: center; --cell-justify: center">No. of inner crosses</td><td style="text-align: center; --cell-justify: center">$0$</td><td style="text-align: center; --cell-justify: center">$1$</td><td style="text-align: center; --cell-justify: center">$3$</td><td style="text-align: center; --cell-justify: center">$6$</td><td style="text-align: center; --cell-justify: center">$\dots$</td><td style="text-align: center; --cell-justify: center">$45$</td><td style="text-align: center; --cell-justify: center">$\dots$</td><td style="text-align: center; --cell-justify: center">$325$</td><td style="text-align: center; --cell-justify: center">$\dots$</td><td style="text-align: center; --cell-justify: center">$w$</td><td style="text-align: center; --cell-justify: center">$\dots$</td><td style="text-align: center; --cell-justify: center">$x$</td></tr></tbody></table></div></div>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Using the pattern in the table, or otherwise, calculate the value of</span></div><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">$u$</span></div></li><li><div class="list-item-content"><span class="list-item-text">$v$</span></div></li><li><div class="list-item-content"><span class="list-item-text">$w$</span></div></li></ol></li><li><div class="list-item-content"><span class="list-item-text">Write down a formula for</span></div><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">$x$ in terms of $n$</span></div></li><li><div class="list-item-content"><span class="list-item-text">$y$ in terms of $n$</span></div></li></ol></li></ol>

The dots in each of the following diagrams form the border of an equilateral triangle which surrounds the inner cross(es) that is/are required to form the triangular pattern. The number of layer of dots of each triangle can be obtained by counting the dots on the side of the triangle.

The table below gives information about the numbers of dots and crosses in triangles of various layers.

Using the pattern in the table, or otherwise, calculate the value of

IP1U7 Exercise Q12

<p>A <strong>series</strong> is the sum of the terms of a sequence. Mathematically, if we have a sequence $a_{1}, a_{2}, a_{3}, \dots, a_{n}$, the corresponding series is written as:</p>
<p style="text-align: center">$S_{n} = a_{1} + a_{2} + a_{3} + \dots a_{n}.$</p>
<p>The series can be either <strong>finite</strong> (with a limited number of terms) or <strong>infinite</strong> (with terms continuing indefinitely).</p>
<p>In general, the general term of a sequence can be expressed as:</p>
<p style="text-align: center">\[T_{n} = S_{n} - S_{n-1}\]</p>

A **series** is the sum of the terms of a sequence. Mathematically, if we have a sequence $a_{1}, a_{2}, a_{3}, \dots, a_{n}$, the corresponding series is written as:

$S_{n} = a_{1} + a_{2} + a_{3} + \dots a_{n}.$

The series can be either **finite** (with a limited number of terms) or **infinite** (with terms continuing indefinitely).

In general, the general term of a sequence can be expressed as:

A Brief Introduction of Series

<p>The first four terms of a sequence are $8a+3b$, $10a+2b$, $12a+b$ and $14a$.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Write down the next two terms.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Express, in terms of $a$, $b$ and $n$, the $n$ th term of the sequence.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">If the $n$ th term of the sequence is $38a-12b$, find the value of $n$.<span class="inline-marks">[2]</span></div></li></ol>

The first four terms of a sequence are $8a+3b$, $10a+2b$, $12a+b$ and $14a$.

Express, in terms of $a$, $b$ and $n$, the $n$th term of the sequence.

If the $n$th term of the sequence is $38a-12b$, find the value of $n$.

IP1 Class Test (May 2025) Q14

<p>In the $1^{st}$ month, a farm has a pair of rabbits. Then in the $3^{rd}$ month, the pair of rabbits grow up and reproduce another pair of rabbits. Suppose that: </p>
<p>1. It always takes two months for rabbits to grow up and be able to reproduce baby rabbits. </p>
<p>2. After growing up, every pair of rabbits reproduce one pair of rabbits every month. </p>
<p>3. No rabbit dies on this farm.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Find the number of pair(s) of rabbits on the farm with respect to time (in month), by completing the following table. (Hint: from the $3^{\text{rd}}$ month onwards, the number of pair(s) of rabbits in each month equals the sum of the number of pair(s) of rabbits in the previous month and the number of pair(s) of new-born rabbits this month.)</span></div></li></ol>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 400px;"><colgroup><col style="width: 150px;"><col style="width: 250px;"></colgroup><thead><tr><th style="text-align: center; --cell-justify: center">Time (Month)</th><th style="text-align: center; --cell-justify: center">Number of Pair(s) of Rabbits, $P_n$</th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center">$1$</td><td style="text-align: center; --cell-justify: center">$1$</td></tr><tr><td style="text-align: center; --cell-justify: center">$2$</td><td style="text-align: center; --cell-justify: center">$1$</td></tr><tr><td style="text-align: center; --cell-justify: center">$3$</td><td style="text-align: center; --cell-justify: center">$2$</td></tr><tr><td style="text-align: center; --cell-justify: center">$4$</td><td style="text-align: center; --cell-justify: center">$3$</td></tr><tr><td style="text-align: center; --cell-justify: center">$5$</td><td style="text-align: center; --cell-justify: center">$5$</td></tr><tr><td style="text-align: center; --cell-justify: center">$6$</td><td style="text-align: center; --cell-justify: center">$8$</td></tr><tr><td style="text-align: center; --cell-justify: center">$7$</td><td style="text-align: left; --cell-justify: flex-start"></td></tr><tr><td style="text-align: center; --cell-justify: center">$8$</td><td style="text-align: left; --cell-justify: flex-start"></td></tr><tr><td style="text-align: center; --cell-justify: center">$9$</td><td style="text-align: left; --cell-justify: flex-start"></td></tr></tbody></table></div></div>
<ol class="i-list" start="3"><li><div class="list-item-content"><span class="list-item-text">State a rule of the number sequence formed by the number of pair(s) of rabbits on the farm, $P_n$.</span></div></li><li><div class="list-item-paragraph">Based on your answers in <strong>(i)</strong> and <strong>(ii)</strong>, Write down the next two terms of the following sequences.</div><ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">$-3, 2, -1, 1, 0, \dots$</span></div></li><li><div class="list-item-content"><span class="list-item-text">$3, 2, 5, 7, 12, \dots$</span></div></li></ol></li></ol>

In the $1^{st}$ month, a farm has a pair of rabbits. Then in the $3^{rd}$ month, the pair of rabbits grow up and reproduce another pair of rabbits. Suppose that: 

1. It always takes two months for rabbits to grow up and be able to reproduce baby rabbits. 

2. After growing up, every pair of rabbits reproduce one pair of rabbits every month. 

Find the number of pair(s) of rabbits on the farm with respect to time (in month), by completing the following table. (Hint: from the $3^{\text{rd}}$ month onwards, the number of pair(s) of rabbits in each month equals the sum of the number of pair(s) of rabbits in the previous month and the number of pair(s) of new-born rabbits this month.)

State a rule of the number sequence formed by the number of pair(s) of rabbits on the farm, $P_n$.

Based on your answers in (i) and (ii), Write down the next two terms of the following sequences.

 Number Patterns

<p>A botanist is monitoring the growth of a certain plant. She recorded the number of leaves and flowers of the plant with respect to time, which is shown in the following tables.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Complete the first table that records the number of leaves and the number of flowers of this plant for the first six weeks.</div><div class="content-block list-item-image" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1767612223419-jnv55i8rvj.png?token=eyJraWQiOiJzdG9yYWdlLXVybC1zaWduaW5nLWtleV9iZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2MTIyMjM0MTktam52NTVpOHJ2ai5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzg1NzEwNTkwLCJleHAiOjE3ODU3MTA4OTB9.4KQlZevsgO0Wa7tPabqnKN2tQLvIRxP68bspykWo5iA" alt="List item image" class="aligncenter custom-width" style="max-width: min(400px, 100%); height: auto;" /></div></li><li><div class="list-item-content"><span class="list-item-text">Based on your answers in (i), find a formula for the number of leaves in the $k^{th}$ week in terms of $k$, where $1 \le k \le 10$.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Hence, find the number of leaves of this plant in week $8$.</span></div></li></ol>
<p>From the $11^{th}$ week, the leaves of the plant start falling, meanwhile the plant starts blooming. Assume the plant's leaf-falling and flower-blooming are at two fixed rates, respectively. On the $m^{th}$ week, all the leaves have fallen.</p>
<ol class="i-list" start="4"><li><div class="list-item-paragraph">Complete the second table that records the number of leaves and the number of flowers of this plant since the $11^{th}$ week.</div><div class="content-block list-item-image" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1767612271915-51pvem951p6.png?token=eyJraWQiOiJzdG9yYWdlLXVybC1zaWduaW5nLWtleV9iZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2MTIyNzE5MTUtNTFwdmVtOTUxcDYucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4NTcxMDU5MCwiZXhwIjoxNzg1NzEwODkwfQ.5U0RY5bHkuJagFLEJgj6EVRtriCj_OoBtLKyFomXpB4" alt="List item image" class="aligncenter custom-width" style="max-width: min(400px, 100%); height: auto;" /></div></li><li><div class="list-item-content"><span class="list-item-text">Based on your answers in (iv), find a formula for the number of flowers in the $n^{th}$ week in terms of $n$, where $n \ge 11$ (i.e. $n$ exceeds $10$).</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the values of $m$ and $p$.</span></div></li></ol>

A botanist is monitoring the growth of a certain plant. She recorded the number of leaves and flowers of the plant with respect to time, which is shown in the following tables.

Complete the first table that records the number of leaves and the number of flowers of this plant for the first six weeks.

Based on your answers in (i), find a formula for the number of leaves in the $k^{th}$ week in terms of $k$, where $1 \le k \le 10$.

Hence, find the number of leaves of this plant in week $8$.

From the $11^{th}$ week, the leaves of the plant start falling, meanwhile the plant starts blooming. Assume the plant's leaf-falling and flower-blooming are at two fixed rates, respectively. On the $m^{th}$ week, all the leaves have fallen.

Complete the second table that records the number of leaves and the number of flowers of this plant since the $11^{th}$ week.

Based on your answers in (iv), find a formula for the number of flowers in the $n^{th}$ week in terms of $n$, where $n \ge 11$ (i.e. $n$ exceeds $10$).

<p>A family of chemical compounds are made up of carbon, hydrogen and oxygen atoms. The number of carbon, hydrogen and oxygen atoms of each of the first four members in the family are shown in the following table.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Complete the table:</div><div class="content-block list-item-image" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1767612092592-1prppp4vbd8.png?token=eyJraWQiOiJzdG9yYWdlLXVybC1zaWduaW5nLWtleV9iZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2MTIwOTI1OTItMXBycHBwNHZiZDgucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4NTcxMDU5MCwiZXhwIjoxNzg1NzEwODkwfQ.r6ugMkiFQUEOyb3X2REBnsuTuZDgfRw_qAFU0lYejLc" alt="List item image" class="aligncenter custom-width" style="max-width: min(400px, 100%); height: auto;" /></div></li><li><div class="list-item-content"><span class="list-item-text">Find a formula that represents the relationship between the number of carbon atoms, $C_n$ and the number of hydrogen atoms, $H_n$ of the $n^{th}$ member in the family.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Suppose the $m^{th}$ member of the family has $50$ hydrogen atoms, find the value of $m$. Hence find the number of carbon atoms, $C_m$ and the number of oxygen atoms, $O_m$ that this member has.</span></div></li></ol>

A family of chemical compounds are made up of carbon, hydrogen and oxygen atoms. The number of carbon, hydrogen and oxygen atoms of each of the first four members in the family are shown in the following table.

Find a formula that represents the relationship between the number of carbon atoms, $C_n$ and the number of hydrogen atoms, $H_n$ of the $n^{th}$ member in the family.

Suppose the $m^{th}$ member of the family has $50$ hydrogen atoms, find the value of $m$. Hence find the number of carbon atoms, $C_m$ and the number of oxygen atoms, $O_m$ that this member has.

<p>Given that the first $5$ terms of a sequence, $T_n$, are $-2, 2, 6, 10, 14, \dots$, where&nbsp;$n = 1, 2, 3, 4$ and $5$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Find a formula for the $n^{th}$ term of the sequence, $T_n$, in terms of $n$.<br>Hence find $T_0$. (Hint: substitute $n = 0$ into the formula of the general term of $T_n$.)</div></li><li><div class="list-item-paragraph">In the grid below, plot $T_n$ against $n$, i.e., draw the points of which the coordinates are in the form of $(n, T_n)$, where&nbsp;$n = 1, 2, 3, 4$ and $5$. Hence, draw a straight line that passes through these points.</div></li><li><div class="list-item-content"><span class="list-item-text"><strong>Using your graph</strong>, find the gradient, $m$, and the $T_n$-intercept, $c$, of the straight line. Hence write down the equation of the straight line in the form of $T_n = mn + c$.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Based on your answers above, determine whether</span></div><ol class="abc-list"><li><div class="list-item-paragraph">whether the point $\left( \frac{1}{2},-4 \right)$ lies on the graph of the straight line or not, and</div></li><li><div class="list-item-content"><span class="list-item-text">$-4$ is a term of the sequence $T_n$ or not. Explain your answers.</span></div></li></ol><div class="list-item-after-content" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1767611033034-otwyfyeja5.png?token=eyJraWQiOiJzdG9yYWdlLXVybC1zaWduaW5nLWtleV9iZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2MTEwMzMwMzQtb3R3eWZ5ZWphNS5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzg1NzEwNTkwLCJleHAiOjE3ODU3MTA4OTB9.KHS-2Jy0P31EiuPEP1MLzhtMXQFGmnPyJJZm_DpT-jo" alt="After sub-items image" class="aligncenter custom-width" style="max-width: min(400px, 100%); height: auto;" /></div></li></ol>

Given that the first $5$ terms of a sequence, $T_n$, are $-2, 2, 6, 10, 14, \dots$, where $n = 1, 2, 3, 4$ and $5$.

Find a formula for the $n^{th}$ term of the sequence, $T_n$, in terms of $n$. Hence find $T_0$. (Hint: substitute $n = 0$ into the formula of the general term of $T_n$.)

On the grid below, plot $T_n$ against $n$, i.e., draw the points of which the coordinates are in the form of $(n, T_n)$, where $n = 1, 2, 3, 4$ and $5$. Hence, draw a straight line that passes through these points.

**Using your graph**, find the gradient, $m$, and the $T_n$-intercept, $c$, of the straight line. Hence write down the equation of the straight line in the form of $T_n = mn + c$.

Based on your answers above, determine whether

<p>Consider the following number pattern:</p>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1767610853882-177f98ezqk7.png?token=eyJraWQiOiJzdG9yYWdlLXVybC1zaWduaW5nLWtleV9iZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2MTA4NTM4ODItMTc3Zjk4ZXpxazcucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4NTcxMDU5MCwiZXhwIjoxNzg1NzEwODkwfQ.Uyd-5aOHp_Gab1ni9EbUnV8M9ry5QqZofwjcbsSNcbE" style="max-width: min(300px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="i-list"><li><div class="list-item-paragraph">State a rule of the numbers at the centre of each row, i.e., $1, 4, 9, 16, 25, \dots$<br>Hence find the number at the centre of the $n^{th}$ row.</div></li><li><div class="list-item-content"><span class="list-item-text">State a rule for each row of numbers in the pattern above.</span></div></li><li><div class="list-item-paragraph">Based on your answers in <strong>(i)</strong> and <strong>(ii)</strong>, complete the following table:</div></li></ol>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 540px;"><colgroup><col style="width: 90px;"><col style="width: 450px;"></colgroup><thead><tr style="height: 35px;"><th style="height: 35px; min-height: 35px; text-align: center; --cell-justify: center">Row Number</th><th style="height: 35px; min-height: 35px; text-align: center; --cell-justify: center">Sum of the Numbers of the Row, $S_n$</th></tr></thead><tbody><tr style="height: 35px;"><td style="height: 35px; min-height: 35px; text-align: center; --cell-justify: center">$1$</td><td style="height: 35px; min-height: 35px; text-align: left; --cell-justify: flex-start"><p>$ \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\;\;\, 1 \quad = 1 \quad\;\;\;\;\; = 1^3$</p></td></tr><tr style="height: 35px;"><td style="height: 35px; min-height: 35px; text-align: center; --cell-justify: center">$2$</td><td style="height: 35px; min-height: 35px; text-align: left; --cell-justify: flex-start"><p>$ \qquad\qquad\qquad\qquad\qquad\qquad\qquad\;\;\;\;\; 2 + 4 + 2 \quad = 8 \quad\;\;\;\;\; = 2^3$</p></td></tr><tr style="height: 35px;"><td style="height: 35px; min-height: 35px; text-align: center; --cell-justify: center">$3$</td><td style="height: 35px; min-height: 35px; text-align: left; --cell-justify: flex-start"><p>$ \qquad\qquad\qquad\qquad\qquad\quad\;\;\;\; 3 + 6 + 9 + 6 + 3 \quad = 27 \quad\;\;\; = 3^3$</p></td></tr><tr style="height: 35px;"><td style="height: 35px; min-height: 35px; text-align: center; --cell-justify: center">$4$</td><td style="height: 35px; min-height: 35px; text-align: left; --cell-justify: flex-start"><p>$ \qquad\qquad\qquad\quad\; 4 + 8 + 12 + 16 + 12 + 8 + 4 \quad = 64 \quad\;\;\; = 4^3$</p></td></tr><tr style="height: 35px;"><td style="height: 35px; min-height: 35px; text-align: center; --cell-justify: center">$5$</td><td style="height: 35px; min-height: 35px; text-align: left; --cell-justify: flex-start"><p>$ \quad\quad 5 + 10 + 15 + 20 + 25 + 20 + 15 + 10 + 5 \quad = 125 \quad\; = 5^3$</p></td></tr><tr style="height: 35px;"><td style="height: 35px; min-height: 35px; text-align: center; --cell-justify: center">$6$</td><td style="height: 35px; min-height: 35px; text-align: center; --cell-justify: center"></td></tr><tr style="height: 35px;"><td style="height: 35px; min-height: 35px; text-align: center; --cell-justify: center">$7$</td><td style="height: 35px; min-height: 35px; text-align: left; --cell-justify: flex-start"></td></tr><tr style="height: 35px;"><td style="height: 35px; min-height: 35px; text-align: center; --cell-justify: center">$\dots$</td><td style="height: 35px; min-height: 35px; text-align: center; --cell-justify: center">$\dots$</td></tr><tr style="height: 35px;"><td style="height: 35px; min-height: 35px; text-align: center; --cell-justify: center">$n$</td><td style="height: 35px; min-height: 35px; text-align: left; --cell-justify: flex-start"><p>$n + 2n + \dots + (n - 1)n + n^2 + (n - 1)n + \dots + 2n + n \qquad = $ ________</p></td></tr></tbody></table></div></div>

State a rule of the numbers at the centre of each row, i.e., $1, 4, 9, 16, 25, \dots$ Hence find the number at the centre of the $n^{th}$ row.

State a rule for each row of numbers in the pattern above.

Based on your answers in (i) and (ii), complete the following table:

<p>Consider the following number pattern:</p>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1770641322976-jrogjksfkrs.png?token=eyJraWQiOiJzdG9yYWdlLXVybC1zaWduaW5nLWtleV9iZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NzA2NDEzMjI5NzYtanJvZ2prc2ZrcnMucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4NTcxMDU5MCwiZXhwIjoxNzg1NzEwODkwfQ.GU2IUnNKg1BQtEicmi9dUPuLX38gJraqb0o9DoGWD2g" style="max-width: min(200px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">State a rule of the number pattern above.</span></div></li><li><div class="list-item-paragraph">Hence complete the following table:</div><div class="table-container"><table style="table-layout: fixed; width: 550px;"><colgroup><col style="width: 100px;"><col style="width: 450px;"></colgroup><thead><tr><th style="text-align: center; --cell-justify: center"><p><strong>Row Number</strong></p></th><th style="text-align: center; --cell-justify: center"><p><strong>Sum of the Numbers of the Row, $S_n$</strong></p></th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center"><p>$1$</p></td><td style="text-align: left; --cell-justify: flex-start"><p>$ \qquad\qquad\qquad\qquad\qquad\qquad\qquad\;\; 1 \quad\; = 1 \quad\; = 2^0$</p></td></tr><tr><td style="text-align: center; --cell-justify: center"><p>$2$</p></td><td style="text-align: left; --cell-justify: flex-start"><p>$ \qquad\qquad\qquad\qquad\qquad\qquad\;\;\; 1 + 1 \quad\; = 2 \quad\; = 2^1$</p></td></tr><tr><td style="text-align: center; --cell-justify: center"><p>$3$</p></td><td style="text-align: left; --cell-justify: flex-start"><p>$ \qquad\qquad\qquad\qquad\qquad\;\;\;\; 1 + 2 + 1 \quad\; = 4 \quad\,\, = 2^2$</p></td></tr><tr><td style="text-align: center; --cell-justify: center"><p>$4$</p></td><td style="text-align: left; --cell-justify: flex-start"><p>$ \qquad\qquad\qquad\qquad\;\;\;\;\; 1 + 3 + 3 + 1 \quad\; = 8 \quad\;\, = 2^3$</p></td></tr><tr><td style="text-align: center; --cell-justify: center"><p>$5$</p></td><td style="text-align: left; --cell-justify: flex-start"><p>$ \qquad\qquad\qquad\;\;\;\;\;\; 1 + 4 + 6 + 4 + 1 \quad\; = 16 \quad = 2^4$</p></td></tr><tr><td style="text-align: center; --cell-justify: center"><p>$6$</p></td><td style="text-align: left; --cell-justify: flex-start"></td></tr><tr><td style="text-align: center; --cell-justify: center"><p>$7$</p></td><td style="text-align: left; --cell-justify: flex-start"></td></tr><tr><td style="text-align: center; --cell-justify: center"><p>...</p></td><td style="text-align: center; --cell-justify: center"><p>...</p></td></tr><tr><td style="text-align: center; --cell-justify: center"><p>$n$</p></td><td style="text-align: center; --cell-justify: center"><p>$1 + (n - 1) + ... + (n - 1) + 1 = $ ________</p></td></tr></tbody></table></div></li></ol>

State a rule of the number pattern above.

<p>A sequence of patterns formed by circles is shown below.</p>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1767610418913-fe229mghgon.png?token=eyJraWQiOiJzdG9yYWdlLXVybC1zaWduaW5nLWtleV9iZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2MTA0MTg5MTMtZmUyMjltZ2hnb24ucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4NTcxMDU5MCwiZXhwIjoxNzg1NzEwODkwfQ.LLR7JvuiEkVPuHc2X1qu144nL9wvD6l72j71Bi8xPwU" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Draw Pattern $5$.</span></div></li><li><div class="list-item-paragraph">Complete the following table:</div><div class="content-block list-item-image" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1767610430635-bm9x2p40qmj.png?token=eyJraWQiOiJzdG9yYWdlLXVybC1zaWduaW5nLWtleV9iZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2MTA0MzA2MzUtYm05eDJwNDBxbWoucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4NTcxMDU5MCwiZXhwIjoxNzg1NzEwODkwfQ.geZClRMFKZpbsKB2uSj_KMGfjqlr_ohWJnbd1A_rTCo" alt="List item image" class="aligncenter custom-width" style="max-width: min(400px, 100%); height: auto;" /></div></li><li><div class="list-item-content"><span class="list-item-text">Based on your answers in (ii), find</span></div><ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">the number of circles in the lowest line and the total number of circles, respectively, in pattern $39$.</span></div></li><li><div class="list-item-content"><span class="list-item-text">the total number of circles, $T_n$, in the pattern where the number of circles in the lowest line, $k$, is $100$.</span></div></li></ol></li></ol>

A sequence of patterns formed by circles is shown below.

<p>A sequence of patterns formed by dots and lines is shown below.</p>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1767610256115-k8l2huimri.png?token=eyJraWQiOiJzdG9yYWdlLXVybC1zaWduaW5nLWtleV9iZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2MTAyNTYxMTUtazhsMmh1aW1yaS5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzg1NzEwNTkwLCJleHAiOjE3ODU3MTA4OTB9.Y9cA0UnqyjLHwU0ckdOX-wBlijF7i3J4aaN574dRVEg" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Draw Pattern $4$.</span></div></li><li><div class="list-item-paragraph">Complete the following table:</div><div class="content-block list-item-image" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1767610265780-i3wk1afriul.png?token=eyJraWQiOiJzdG9yYWdlLXVybC1zaWduaW5nLWtleV9iZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2MTAyNjU3ODAtaTN3azFhZnJpdWwucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4NTcxMDU5MCwiZXhwIjoxNzg1NzEwODkwfQ.K40QcKXpJkgLjnfAK8bQKLj2GO5a57EOJ15R-0eUmUU" alt="List item image" class="aligncenter custom-width" style="max-width: min(400px, 100%); height: auto;" /></div></li><li><div class="list-item-content"><span class="list-item-text">Based on your answers in (ii), find</span></div><ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">the number of dots and lines, respectively, to make pattern $50$.</span></div></li><li><div class="list-item-content"><span class="list-item-text">the pattern number that uses a total of $2024$ dots. Hence find the number of lines in this pattern.</span></div></li></ol></li><li><div class="list-item-content"><span class="list-item-text">Find a formula that represents the relationship between the number of dots, $D$, the number of lines, $L$, and the pattern number, $n$, in the sequence shown above.</span></div></li></ol>

A sequence of patterns formed by dots and lines is shown below.

Find a formula that represents the relationship between the number of dots, $D$, the number of lines, $L$, and the pattern number, $n$, in the sequence shown above.

<p>A sequence of patterns formed by squares is shown below.</p>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1767610085953-1v26wh9oli6.png?token=eyJraWQiOiJzdG9yYWdlLXVybC1zaWduaW5nLWtleV9iZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2MTAwODU5NTMtMXYyNndoOW9saTYucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4NTcxMDU5MCwiZXhwIjoxNzg1NzEwODkwfQ.hpu8GePo-ZflrLieJnAv4qmTs_0ffapjxxslH29Wmjc" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Draw Pattern $4$.</span></div></li><li><div class="list-item-paragraph">Complete the following table:</div><div class="content-block list-item-image" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1767610104605-vr1zeladf9j.png?token=eyJraWQiOiJzdG9yYWdlLXVybC1zaWduaW5nLWtleV9iZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2MTAxMDQ2MDUtdnIxemVsYWRmOWoucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4NTcxMDU5MCwiZXhwIjoxNzg1NzEwODkwfQ.vjwUb0jUo-WG6EbdGfXp7p5kMkDX5jHsgvIpXWokHzs" alt="List item image" class="aligncenter custom-width" style="max-width: min(280px, 100%); height: auto;" /></div></li><li><div class="list-item-content"><span class="list-item-text">Based on your answers in (ii), find</span></div><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">the number of squares to make pattern $37$.</span></div></li><li><div class="list-item-content"><span class="list-item-text">the pattern number that uses a total of $2023$ squares.</span></div></li></ol></li><li><div class="list-item-content"><span class="list-item-text">Is there a pattern that is made up of $242$ squares? Explain your answer.</span></div></li></ol>

A sequence of patterns formed by squares is shown below.

Is there a pattern that is made up of $242$ squares? Explain your answer.

<p>Consider the following number pattern: \[1^2 - 0^2 = 1\] \[2^2 - 1^2 = 3\] \[3^2 - 2^2 = 5\] \[\dots\] \[m^2 - (m - 1)^2 = 65\] \[\dots\]</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Write down the $5^{th}$ and the $6^{th}$ row of the number pattern.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Write down, in terms of $n$, the equation that appears in the $n^{th}$ row of the pattern.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the integer value of $m$.</span></div></li></ol>

Consider the following number pattern: $$1^2 - 0^2 = 1$$ $$2^2 - 1^2 = 3$$ $$3^2 - 2^2 = 5$$ $$\dots$$ $$m^2 - (m - 1)^2 = 65$$ $$\dots$$

Write down the $5^{th}$ and the $6^{th}$ row of the number pattern.

Write down, in terms of $n$, the equation that appears in the $n^{th}$ row of the pattern.

<p>Consider the following number pattern:</p>
<p>$ \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\;\;\;\; 1 \quad = 1 \quad\,\,\,\,\,\, = 1^2$</p>
<p>$ \qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\quad\quad\quad\quad\;\, 1 + 3 \quad = 4 \quad\,\,\,\,\,\, = 2^2$</p>
<p>$ \qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\quad\quad\;\;\; 1 + 3 + 5 \quad = 9 \quad\,\,\,\,\,\, = 3^2$</p>
<p>$ \qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\quad\, 1 + 3 + 5 + 7 \quad = 16 \quad\,\,\, = 4^2$</p>
<p style="text-align: center">$...$</p>
<p>$ \qquad\qquad\qquad\qquad\qquad\qquad\qquad\; 1 + 3 + 5 + ... + k \quad = 289 \quad =$</p>
<p style="text-align: center">$...$</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Write down the $5^{th}$ and the $6^{th}$ row of the number pattern.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Fill in the blank in the $k^{th}$ row and find the value of $k$.</span></div></li><li><div class="list-item-paragraph">Find a formula for the ${n^{\mathrm{th}}}$ row of the pattern in terms of ${n}$ .</div></li></ol>

$ \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\;\;\;\; 1 \quad = 1 \quad\,\,\,\,\,\, = 1^2$

$ \qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\quad\quad\quad\quad\;\, 1 + 3 \quad = 4 \quad\,\,\,\,\,\, = 2^2$

$ \qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\quad\quad\;\;\; 1 + 3 + 5 \quad = 9 \quad\,\,\,\,\,\, = 3^2$

$ \qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\quad\, 1 + 3 + 5 + 7 \quad = 16 \quad\,\,\, = 4^2$

$ \qquad\qquad\qquad\qquad\qquad\qquad\qquad\; 1 + 3 + 5 + ... + k \quad = 289 \quad =$

Fill in the blank in the $k^{th}$ row and find the value of $k$.

Find, in terms of $n$, the formula for $T_n$ in the $n^{th}$ row of the pattern.

<p style="text-align: center">Consider the following number pattern: \[\frac{1}{2} = \frac{1}{1 \times 2} = 1 - \frac{1}{2}\] \[\frac{1}{6} = \frac{1}{2 \times 3} = \frac{1}{2} - \frac{1}{3}\] \[\frac{1}{12} = \frac{1}{3 \times 4} = \frac{1}{3} - \frac{1}{4}\] \[\frac{1}{20} = \frac{1}{4 \times 5} = \frac{1}{4} - \frac{1}{5}\] \[\dots\] \[\frac{1}{110} = \frac{1}{m \times k} = \frac{1}{m} - \frac{1}{k}\] \[\dots\]</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Write down the $5^{th}$ and the $6^{th}$ row of the number pattern.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the integer values of $m$ and $k$.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find a formula for the $n^{th}$ row of the pattern in terms of $n$.</span></div></li></ol>

Consider the following number pattern: $$\frac{1}{2} = \frac{1}{1 \times 2} = 1 - \frac{1}{2}$$ $$\frac{1}{6} = \frac{1}{2 \times 3} = \frac{1}{2} - \frac{1}{3}$$ $$\frac{1}{12} = \frac{1}{3 \times 4} = \frac{1}{3} - \frac{1}{4}$$ $$\frac{1}{20} = \frac{1}{4 \times 5} = \frac{1}{4} - \frac{1}{5}$$ $$\dots$$ $$\frac{1}{110} = \frac{1}{m \times k} = \frac{1}{m} - \frac{1}{k}$$ $$\dots$$

Find a formula for the $n^{th}$ row of the pattern in terms of $n$.

<p>Consider the following number pattern:\[12 = 3 \times 4\]\[20 = 4 \times 5\]\[30 = 5 \times 6\]\[42 = 6 \times 7\]\[\dots\]\[342 = (t + 2)(t + 3)\]\[\dots\]</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Write down the $5^{th}$ and the $6^{th}$ row of the number pattern.</span></div></li><li><div class="list-item-content"><span class="list-item-text">By finding the integer value of $t$, determine which row does $342$ appear.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find a formula for the $n^{th}$ row of the pattern in terms of $n$.</span></div></li></ol>

Consider the following number pattern:$$12 = 3 \times 4$$$$20 = 4 \times 5$$$$30 = 5 \times 6$$$$42 = 6 \times 7$$$$\dots$$$$342 = (t + 2)(t + 3)$$$$\dots$$

By finding the integer value of $t$, determine which row does $342$ appear.

<p>For the following number sequence: \[\frac{1}{2}, \frac{3}{7}, \frac{5}{12}, \frac{7}{17}, \frac{9}{22}, \dots\]</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Write down the next two terms of the sequence.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find a formula for the $n^{th}$ term of the sequence in terms of $n$.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Hence find the $16^{th}$ term.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the lowest common multiple of the denominators of the $7^{th}$ and the $11^{th}$ term. Hence, find the difference between the $7^{th}$ and the $11^{th}$ term.</span></div></li></ol>

For the following number sequence: $$\frac{1}{2}, \frac{3}{7}, \frac{5}{12}, \frac{7}{17}, \frac{9}{22}, \dots$$

Write down the next two terms of the sequence.

Find a formula for the $n^{th}$ term of the sequence in terms of $n$.

Find the lowest common multiple of the denominators of the $7^{th}$ and the $11^{th}$ term. Hence, find the difference between the $7^{th}$ and the $11^{th}$ term.

<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Given that the $n^{th}$ term, $T_n$, of a sequence is $T_n = n^2 + 1$.</span></div><ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Write down the first $5$ terms of the sequence.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the lowest common multiple of the $7^{th}$ and the $8^{th}$ term.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the highest common factor of the $4^{th}$ and the $13^{th}$ term.</span></div></li></ol></li><li><div class="list-item-content"><span class="list-item-text">The first $5$ terms of another sequence are $4, 7, 12, 19, 28$.</span></div><ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">By comparing this sequence and the sequence in (i), find a formula for the $n^{th}$ term of this sequence in terms of $n$.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Hence find the $21^{st}$ term of this sequence.</span></div></li></ol></li></ol>

Given that the $n^{th}$ term, $T_n$, of a sequence is $T_n = n^2 + 1$.

The first $5$ terms of another sequence are $4, 7, 12, 19, 28$.

<p>Write down the next two terms of each of the following sequences.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">$0, 3, 8, 15, 24, \dots$</span></div></li><li><div class="list-item-content"><span class="list-item-text">$2, 3, -1, 0, -4, \dots$</span></div></li><li><div class="list-item-content"><span class="list-item-text">$200, 201, 196, 197, 192, \dots$</span></div></li><li><div class="list-item-content"><span class="list-item-text">$1, 3, 7, 15, 31, \dots$</span></div></li></ol>

Write down the next two terms of each of the following sequences.

Math Question Bank