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PSLE

PSLE Foundation Mathematics

<p>The first four terms in a sequence are $1, 4, 9$ and $16$.</p>
<ol class="abc-list"><li data-list-item-id="item-1785756533528-2co5fuelw" data-list-item-path="item-1785756533528-2co5fuelw"><div class="list-item-paragraph"><p>Write down the $6$th term of this sequence.</p><span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1785756534704-9z7mggxiy" data-list-item-path="item-1785756534704-9z7mggxiy"><div class="list-item-paragraph"><p>Find an expression, in terms of $n$, for the $n$th term, $T_n$, of the sequence.</p><span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1785756630861-b8kpl7gee" data-list-item-path="item-1785756630861-b8kpl7gee"><div class="list-item-paragraph"><p>Is the number $99$ part of the sequence above? Explain your answer.</p><span class="inline-marks">[1]</span></div></li></ol>

The first four terms in a sequence are $1, 4, 9$ and $16$.

General Term of a Sequence

Testing Sequence Membership

2022 AMKSS EOY P1 Q3

Number Patterns

<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}$

2022 TKSS EOY P2 Q3

<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 $14a$.

IP1 Class Test (May 2025) Q14

<p>In the figure below, $MON$ is a straight line. $OP$ is a random ray with endpoint $O$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">The rays $OQ_1$ and $OR_1$ are the bisectors of the angle $MOP$ and angle $NOP$, respectively. Find the size of angle $Q_1OR_1$.</div><div class="content-block list-item-image" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/migrated/2024/06/Extra-Practice-Q9-i-diagram.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvbWlncmF0ZWQvMjAyNC8wNi9FeHRyYS1QcmFjdGljZS1ROS1pLWRpYWdyYW0ucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mjc2MzczMCwiZXhwIjoxNzkyNzY0MDMwfQ.e7BBU9XNHDcK68TnQPzFGx8cmGoVhGypExeAIW2UXow" alt="List item image" class="aligncenter custom-width" style="max-width: min(300px, 100%); height: auto;" /></div></li><li><div class="list-item-paragraph">By extending the figure in (i), the rays $OQ_2$ and $OR_2$ are the bisectors of angle $Q_1OP$ and angle $R_1OP$, respectively. Find the size of angle $Q_2OR_2$ and state a pattern or assumption you find about angle $Q_2OR_2$ and angle $Q_nOR_n$ where $n &gt; 2$.</div><div class="content-block list-item-image" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/migrated/2024/06/Extra-Practice-Q9-ii-diagram-1.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvbWlncmF0ZWQvMjAyNC8wNi9FeHRyYS1QcmFjdGljZS1ROS1paS1kaWFncmFtLTEucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mjc2MzczMCwiZXhwIjoxNzkyNzY0MDMwfQ.uKWZvf1omKs9KOnDNowZ38OM0q26NwGM2Zgay-sxYdg" alt="List item image" class="aligncenter custom-width" style="max-width: min(300px, 100%); height: auto;" /></div></li><li><div class="list-item-paragraph"><strong>By extending the figure in (ii)</strong>, the rays $OQ_3$ and $OR_3$ are the bisectors of angle $Q_2OP$ and angle $R_2OP$, respectively. Find the size of angle $Q_3OR_3$ and test whether your assumption also holds for angle $Q_3OR_3$.</div><div class="content-block list-item-image" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/migrated/2024/06/Extra-Practice-Q9-iii-diagram-624x283.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvbWlncmF0ZWQvMjAyNC8wNi9FeHRyYS1QcmFjdGljZS1ROS1paWktZGlhZ3JhbS02MjR4MjgzLnBuZyIsInNjb3BlIjoiZG93bmxvYWQiLCJpYXQiOjE3OTI3NjM3MzAsImV4cCI6MTc5Mjc2NDAzMH0.--CaQ8bAHDV5S0cT-2kzbw9POzChlCqb_Ekk2YfV1tQ" alt="List item image" class="aligncenter custom-width" style="max-width: min(350px, 100%); height: auto;" /></div></li><li><div class="list-item-paragraph"><strong>Based on the pattern you find</strong>, complete the following table, where the <strong>4th</strong> and <strong>5th</strong> patterns are given.</div></li></ol>
<div class="content-block content-block--table"><div class="table-container"><table><thead><tr><th style="text-align: left; --cell-justify: flex-start">Pattern Number, $n$</th><th style="text-align: left; --cell-justify: flex-start">Size of angle $Q_nOR_n$</th></tr></thead><tbody><tr><td style="text-align: left; --cell-justify: flex-start">1</td><td style="text-align: left; --cell-justify: flex-start"></td></tr><tr><td style="text-align: left; --cell-justify: flex-start">2</td><td style="text-align: left; --cell-justify: flex-start"></td></tr><tr><td style="text-align: left; --cell-justify: flex-start">3</td><td style="text-align: left; --cell-justify: flex-start"></td></tr><tr><td style="text-align: left; --cell-justify: flex-start">4</td><td style="text-align: left; --cell-justify: flex-start">$11.25^\circ = 180^\circ \times (\frac{1}{2})^4$</td></tr><tr><td style="text-align: left; --cell-justify: flex-start">5</td><td style="text-align: left; --cell-justify: flex-start">$5.625^\circ = 180^\circ \times (\frac{1}{2})^5$</td></tr><tr><td style="text-align: left; --cell-justify: flex-start">...</td><td style="text-align: left; --cell-justify: flex-start">...</td></tr></tbody></table></div></div>
<ol class="i-list" start="5"><li><div class="list-item-paragraph"><strong>Based on your previous answers</strong>, find the general term of size of $Q_nOR_n$ in terms of $n$, where $OQ_n$, $OR_n$ are the bisectors of angle $Q_{n-1}OP$ and angle $R_{n-1}OP$ respectively and $n$ is an integer that is greater than 3.</div><div class="content-block list-item-image" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/migrated/2024/06/Extra-Practice-Q9-v-diagram-624x266.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvbWlncmF0ZWQvMjAyNC8wNi9FeHRyYS1QcmFjdGljZS1ROS12LWRpYWdyYW0tNjI0eDI2Ni5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkyNzYzNzMwLCJleHAiOjE3OTI3NjQwMzB9.cnlN0qHi7cLK8IOS-3jNgVSSzrjtmOKPsL6clKjeTYc" alt="List item image" class="aligncenter custom-width" style="max-width: min(350px, 100%); height: auto;" /></div></li></ol>

In the figure below, $MON$ is a straight line. $OP$ is a random ray with endpoint $O$.

The rays $OQ_1$ and $OR_1$ are the bisectors of the angle $MOP$ and angle $NOP$, respectively. Find the size of angle $Q_1OR_1$.

By extending the figure in (i), the rays $OQ_2$ and $OR_2$ are the bisectors of angle $Q_1OP$ and angle $R_1OP$, respectively. Find the size of angle $Q_2OR_2$ and state a pattern or assumption you find about angle $Q_2OR_2$ and angle $Q_nOR_n$ where $n > 2$.

Rules of Sequences

Angles

Basic Geometry

<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 data-list-item-id="item-i" data-list-item-path="item-i"><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=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2MTIyMjM0MTktam52NTVpOHJ2ai5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkyNzYzNzMwLCJleHAiOjE3OTI3NjQwMzB9.e8NrcENDIxsWMmVyUr8K5uql_FQ8Q__jyslNLC1SdwM" alt="List item image" class="aligncenter custom-width" style="max-width: min(400px, 100%); height: auto;" /></div></li><li data-list-item-id="item-ii" data-list-item-path="item-ii"><div class="list-item-paragraph">Based on your answers in (i), find a formula for the number of leaves in the $k^{th}$ week in terms of $k$, where&nbsp;$1 \le k \le 10$.</div></li><li data-list-item-id="item-iii" data-list-item-path="item-iii"><div class="list-item-paragraph">Hence, find the number of leaves of this plant in week $8$.</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 data-list-item-id="item-iv" data-list-item-path="item-iv"><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=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2MTIyNzE5MTUtNTFwdmVtOTUxcDYucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mjc2MzczMCwiZXhwIjoxNzkyNzY0MDMwfQ.ZRhXJsnhYWqqXXEvv7aVyBtZb8pUr8xd5W0xze_IemQ" alt="List item image" class="aligncenter custom-width" style="max-width: min(400px, 100%); height: auto;" /></div></li><li data-list-item-id="item-v" data-list-item-path="item-v"><div class="list-item-paragraph">Based on your answers in (iv), find a formula for the number of flowers in the $n^{th}$ week in terms of $n$, where&nbsp;$n \ge 11$ (i.e. $n$ exceeds $10$).</div></li><li data-list-item-id="item-vi" data-list-item-path="item-vi"><div class="list-item-paragraph">Find the values of $m$ and $p$.</div></li></ol>
<p>Source clarification: the first rule gives 99 leaves at week 10, while the second table starts at 120 leaves in week 11. Treat week 11 as the starting observation for the later model; the tabulated fall of 10 leaves per week begins after that observation.</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.

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 $10$).

Source clarification: the first rule gives 99 leaves at week 10, while the second table starts at 120 leaves in week 11. Treat week 11 as the starting observation for the later model; the tabulated fall of 10 leaves per week begins after that observation.

 Number Patterns

<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=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2MTIwOTI1OTItMXBycHBwNHZiZDgucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mjc2MzczMCwiZXhwIjoxNzkyNzY0MDMwfQ.qkkX03eKS4kU38PvxEHrG6rvBEoRGZTet_GV9yEU4Wc" 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 data-list-item-id="item-i" data-list-item-path="item-i"><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 data-list-item-id="item-ii" data-list-item-path="item-ii"><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 data-list-item-id="item-iii" data-list-item-path="item-iii"><div class="list-item-paragraph"><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 $T_n = mn + c$.</div></li><li data-list-item-id="item-iv" data-list-item-path="item-iv"><div class="list-item-paragraph">Based on your answers above, determine whether</div><ol class="abc-list"><li data-list-item-id="item-iva" data-list-item-path="item-iv/item-iva"><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 data-list-item-id="item-ivb" data-list-item-path="item-iv/item-ivb"><div class="list-item-paragraph">$-4$ is a term of the sequence $T_n$ or not. Explain your answers.</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=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2MTEwMzMwMzQtb3R3eWZ5ZWphNS5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkyNzYzNzMwLCJleHAiOjE3OTI3NjQwMzB9.1xEZ_HmBJyE32zorkEhd7ll8yc8TsFN4oIr6fZPlC6s" 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 $5$.

**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

Gradient of a Straight Line

Plotting of Straight Line on Graph Paper

Equation of a Straight Line

Linear Graphs and Its Applications

<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=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2MTA4NTM4ODItMTc3Zjk4ZXpxazcucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mjc2MzczMCwiZXhwIjoxNzkyNzY0MDMwfQ.g__CbJjeG8DkKXEwhizIxQ1_nAJzkKdHuxMi3s6DY5A" style="max-width: min(300px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="i-list"><li data-list-item-id="item-i" data-list-item-path="item-i"><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 data-list-item-id="item-ii" data-list-item-path="item-ii"><div class="list-item-paragraph">State a rule for each row of numbers in the pattern above.</div></li><li data-list-item-id="item-iii" data-list-item-path="item-iii"><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: 16.6667%;"><col style="width: 83.3333%;"></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: center; --cell-justify: center"><p>$   1   = 1   = 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: center; --cell-justify: center"><p>$   2 + 4 + 2   = 8   = 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: center; --cell-justify: center"><p>$   3 + 6 + 9 + 6 + 3   = 27   = 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: center; --cell-justify: center"><p>$   4 + 8 + 12 + 16 + 12 + 8 + 4   = 64   = 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: center; --cell-justify: center"><p>$   5 + 10 + 15 + 20 + 25 + 20 + 15 + 10 + 5   = 125   = 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: center; --cell-justify: center"></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: center; --cell-justify: center">$2n(1+2+\cdots+(n-1))+n^2=$ ________</td></tr></tbody></table></div></div>

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

Arithmetic Progression

<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=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NzA2NDEzMjI5NzYtanJvZ2prc2ZrcnMucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mjc2MzczMCwiZXhwIjoxNzkyNzY0MDMwfQ.xbxLRrlUyLdSdUouWi16yPUt2aurPQacgDpGz32w9-A" style="max-width: min(200px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="i-list"><li data-list-item-id="item-i" data-list-item-path="item-i"><div class="list-item-paragraph">State a rule of the number pattern above.</div></li><li data-list-item-id="item-ii" data-list-item-path="item-ii"><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: 18.1818%;"><col style="width: 81.8182%;"></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: center; --cell-justify: center"><p>$   1   = 1   = 2^0$</p></td></tr><tr><td style="text-align: center; --cell-justify: center"><p>$2$</p></td><td style="text-align: center; --cell-justify: center"><p>$   1 + 1   = 2   = 2^1$</p></td></tr><tr><td style="text-align: center; --cell-justify: center"><p>$3$</p></td><td style="text-align: center; --cell-justify: center"><p>$   1 + 2 + 1   = 4   = 2^2$</p></td></tr><tr><td style="text-align: center; --cell-justify: center"><p>$4$</p></td><td style="text-align: center; --cell-justify: center"><p>$   1 + 3 + 3 + 1   = 8   = 2^3$</p></td></tr><tr><td style="text-align: center; --cell-justify: center"><p>$5$</p></td><td style="text-align: center; --cell-justify: center"><p>$   1 + 4 + 6 + 4 + 1   = 16   = 2^4$</p></td></tr><tr><td style="text-align: center; --cell-justify: center"><p>$6$</p></td><td style="text-align: center; --cell-justify: center"></td></tr><tr><td style="text-align: center; --cell-justify: center"><p>$7$</p></td><td style="text-align: center; --cell-justify: center"></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.

Binomial Coefficients and Pascal's Triangle

Binomial Theorem

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

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

Fill in the blank in the row containing $k$ and find the value of $k$.

<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.

Lowest Common Multiple and Its Applications

Primes, HCF and LCM

<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 $4, 7, 12, 19, 28$.

Highest Common Factor and Its Applications

<p>The first four terms of a sequence are $2, -2, -6, -10, \dots$</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Write down the $6^{th}$ term 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 $150^{th}$ term.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Is $-60$ a term in the sequence? Justify your answer.</span></div></li></ol>

The first four terms of a sequence are $2, -2, -6, -10, \dots$

Write down the $6^{th}$ term of the sequence.

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

Is $-60$ a term in the sequence? Justify your answer.

 General Term of a Sequence

<p>The first four terms of a sequence are $-13, -7, -1, 5$.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Write down the $6^{th}$ term of the sequence.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find, in terms of $n$, a formula for the $n^{th}$ term of the sequence.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Hence find the $100^{th}$ term.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Is $41$ a term in the sequence? Explain your answer.</span></div></li></ol>

The first four terms of a sequence are $-13, -7, -1, 5$.

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

Is $41$ a term in the sequence? Explain your answer.

<p>Given that the $n^{th}$ term, $T_n$, of a sequence is $T_n = 56 - 4n$, find</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">the $3^{rd}$ term,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the difference between the $15^{th}$ and the $5^{th}$ term,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the lowest common multiple of the $10^{th}$ and the $11^{th}$ term.</span></div></li></ol>

Given that the $n^{th}$ term, $T_n$, of a sequence is $T_n = 56 - 4n$, find

the difference between the $15^{th}$ and the $5^{th}$ term,

the lowest common multiple of the $10^{th}$ and the $11^{th}$ term.

<p>Given that the $n^{th}$ term, $T_n$, of a sequence is $T_n = 7n + 5$, find</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">the $2^{nd}$ term,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the sum of the $24^{th}$ and the $27^{th}$ term,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the highest common factor of the $10^{th}$ and the $15^{th}$ term.</span></div></li></ol>

Given that the $n^{th}$ term, $T_n$, of a sequence is $T_n = 7n + 5$, find

the sum of the $24^{th}$ and the $27^{th}$ term,

the highest common factor of the $10^{th}$ and the $15^{th}$ term.

<p>The first five terms of a number sequence are $-10$, $-6$, $-2$, $2$ and $6$.</p>
<ol class="abc-list"><li data-list-item-id="item-1765268359929-7hncax561" data-list-item-path="item-1765268359929-7hncax561"><div class="list-item-paragraph">Write down the $8^{\mathrm{th}}$ term.</div></li><li data-list-item-id="item-1765268364720-fcqmfp48h" data-list-item-path="item-1765268364720-fcqmfp48h"><div class="list-item-paragraph">Find the expression for the $n$th term of the sequence.</div></li><li data-list-item-id="item-1765268371050-xzg32v5ju" data-list-item-path="item-1765268371050-xzg32v5ju"><div class="list-item-paragraph">Find the term in the sequence when the number is $458$.</div></li><li data-list-item-id="item-1765268376151-0n64zk7md" data-list-item-path="item-1765268376151-0n64zk7md"><div class="list-item-paragraph">Is $1107$ a number in the number sequence? Explain your answer.</div></li></ol>
<p>Continue this as an arithmetic sequence.</p>

The first five terms of a number sequence are $-10$, $-6$, $-2$, $2$ and $6$.

Math Question Bank