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PSLE

PSLE Foundation Mathematics

<ol class="abc-list"><li data-list-item-id="item-1760418785015-drvtt3thx" data-list-item-path="item-1760418785015-drvtt3thx"><div class="list-item-paragraph">An infinite geometric progression has first term $a$ and common ratio $r$, where&nbsp;$a$ and $r$ are non-zero. The sum of all the terms after the $n$th term of the progression is equal to twice the $n$th term. Show that the sum to infinity of the progression is three times the first term.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1760418785705-h0df4haho" data-list-item-path="item-1760418785705-h0df4haho"><div class="list-item-paragraph">The positive integers, starting at $~1$, are grouped into sets, as follows.</div><div class="list-item-paragraph" style="text-align: center">$\left\{ 1 \right\}$, $\left\{ 2,3 \right\}$, $\left\{ 4,5,6 \right\}$,…</div><div class="list-item-paragraph">where there are $r$ integers in the $r$th set.</div><ol class="i-list"><li data-list-item-id="subitem-1760418808813-46rddh60u" data-list-item-path="item-1760418785705-h0df4haho/subitem-1760418808813-46rddh60u"><div class="list-item-paragraph">Find, in terms of $r$, the first integer and the last integer in the $r$th set.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="subitem-1760418810028-0vgiv0zq0" data-list-item-path="item-1760418785705-h0df4haho/subitem-1760418810028-0vgiv0zq0"><div class="list-item-paragraph">Prove that the sum of the integers in the $r$th set is $\frac{1}{2}r\left( 1+{{r}^{2}} \right)$.<span class="inline-marks">[2]</span></div></li></ol></li></ol>

Grouped sets

Sum to infinity

2023 YIJC P1 Q5

Sequence and Series

<p>The sum of the first $n$ terms of a series, ${{S}_{n}}$, is given by</p>
<p style="text-align: center">${{S}_{n}}=\frac{1}{a}\left[ 1-{{(a-1)}^{n}} \right]$, where $a$ is a constant and $a\ne 1$, $n\in {{\mathbb{Z}}^{+}}$</p>
<p>Obtain an expression for the ${{n}^{\mathrm{th}}}$ term of the series, ${{T}_{n}}$, and prove that ${{T}_{n}}$ is a geometric series.</p><p style="text-align: right;">[3]</p>
<p>If the sequence ${{T}_{n}}$ is now grouped as follows: $({{T}_{1}}),\ ({{T}_{2}},\ {{T}_{3}},\ {{T}_{4}}),\ ({{T}_{5}},\ {{T}_{6}},\ {{T}_{7}},\ {{T}_{8}},\ {{T}_{9}}),\ \ldots $ where each subsequent bracket has $2$ terms more than the previous bracket, find</p>
<ol class="i-list"><li><div class="list-item-paragraph">the total number of terms in the first $n$ brackets.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">the middle term of the ${{11}^{\mathrm{th}}}$ bracket in terms of $a$.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">the range of values of $a$ for the sum to infinity of the series to exist.</div><div class="list-item-paragraph">Hence, find the least value of $n$ for the sum of all the terms in the first $n$ brackets to be within $0.1\%$ of the sum to infinity of the series when $a=\frac{39}{20}$.</div><p style="text-align: right;">[4]</p></li></ol>

The sum of the first $n$ terms of a series, ${{S}_{n}}$, is given by

${{S}_{n}}=\frac{1}{a}\left[ 1-{{(a-1)}^{n}} \right]$, where $a$ is a constant and $a\ne 1$, $n\in {{\mathbb{Z}}^{+}}$

Obtain an expression for the ${{n}^{\mathrm{th}}}$ term of the series, ${{T}_{n}}$, and prove that ${{T}_{n}}$ is a geometric series.

If the sequence ${{T}_{n}}$ is now grouped as follows: $({{T}_{1}}),\ ({{T}_{2}},\ {{T}_{3}},\ {{T}_{4}}),\ ({{T}_{5}},\ {{T}_{6}},\ {{T}_{7}},\ {{T}_{8}},\ {{T}_{9}}),\ \ldots $ where each subsequent bracket has $2$ terms more than the previous bracket, find

Problems involving inequalities

Proving the progression

Geometric Progression

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

2011 AJC P1 Q12

<p>Part of an isosceles triangle, formed by numbers following a particular pattern 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/converted/1760188119278-yq90ry5idk.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NjAxODgxMTkyNzgteXE5MHJ5NWlkay5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkyNzYzNTkzLCJleHAiOjE3OTI3NjM4OTN9.Q10AnOg8PzpyzGWMWd4PHFRFSmWY51ZD8CyKjpI_p1E" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="i-list"><li data-list-item-id="item-1760188122134-ltw0m8680" data-list-item-path="item-1760188122134-ltw0m8680"><div class="list-item-paragraph">State the middle term in the $n^{\mathrm{th}}$ row.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1760188122742-rz5am31co" data-list-item-path="item-1760188122742-rz5am31co"><div class="list-item-paragraph">Find the first and last term of the $n^{\mathrm{th}}$ row.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1760188124226-a9dxrdbxn" data-list-item-path="item-1760188124226-a9dxrdbxn"><div class="list-item-paragraph">Show that the sum of the terms in the $n^{\mathrm{th}}$ row is $\left( 2n-1 \right){{3}^{n-1}}$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1760188124783-3w7b97ljs" data-list-item-path="item-1760188124783-3w7b97ljs"><div class="list-item-paragraph">Find the minimum number of rows in the triangle for the number of terms in the triangle to exceed $500$.<span class="inline-marks">[3]</span></div></li></ol>

Part of an isosceles triangle, formed by numbers following a particular pattern is shown below:

Arithmetic Progression

2010 AJC P1 Q11

<ol class="abc-list" start="2"><li data-list-item-id="item-1759914519266-13eiuuy5a" data-list-item-path="item-1759914519266-13eiuuy5a"><div class="list-item-paragraph">The arithmetic progression ${2}$, ${4}$, ${6}$, ${8}$, ${10}$, ${12}$, ${...}$ is arranged in rows in the following way:<br>1st Row:               $ \qquad\qquad\; {2}$<br>2nd Row:             $ \qquad\quad\;\;\; {4}$, ${6}$<br>3rd Row:               $ \qquad\qquad {8}$, ${10}$, ${12}$<br>4th Row:               $ \qquad\qquad {14}$, ${16}$, ${18}$, ${20}$<br>${...}$</div><ol class="i-list"><li data-list-item-id="subitem-1759914540764-1b25mxjcd" data-list-item-path="item-1759914519266-13eiuuy5a/subitem-1759914540764-1b25mxjcd"><div class="list-item-paragraph">Show that the first term in the $n^{\mathrm{th}}$ row is ${{n}^{2}}-n+2$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="subitem-1759914542533-kx34btz60" data-list-item-path="item-1759914519266-13eiuuy5a/subitem-1759914542533-kx34btz60"><div class="list-item-paragraph">Find the sum of all the terms from the ${{1}^{\mathrm{st}}}$ row to the $\left( n-1 \right)^{\mathrm{th}}$ row.<span class="inline-marks">[3]</span></div></li></ol></li></ol>

2009 DHS P1 Q9

<p>An arithmetic progression $\left\{ {{u}_{n}} \right\}$ is grouped into sets containing $1$, $3$, $9$, … shown below, so that the number of integers in the $n$$^{\mathrm{th}}$ set is ${{3}^{n-1}}$.</p>
<p>$\left\{ 1 \right\}$, $\left\{ 5,\mathrm{ }9,\mathrm{ }13 \right\}$, $\left\{ 17,\mathrm{ }21,\mathrm{ }25,\mathrm{ }29,\mathrm{ }33,\mathrm{ }37,\mathrm{ }41,\mathrm{ }45,\mathrm{ }49 \right\}$, …</p>
<ol class="i-list"><li data-list-item-id="item-1759839946379-itfd5bh6t" data-list-item-path="item-1759839946379-itfd5bh6t"><div class="list-item-paragraph">In either order, find expressions in terms of $n$, for</div><ol class="abc-list"><li data-list-item-id="subitem-1759839951833-ibppfsmc8" data-list-item-path="item-1759839946379-itfd5bh6t/subitem-1759839951833-ibppfsmc8"><div class="list-item-paragraph">the first integer in the $n$$^{\mathrm{th}}$ set,<span class="inline-marks">[3]</span></div></li><li data-list-item-id="subitem-1759839952709-wfiuz8f3e" data-list-item-path="item-1759839946379-itfd5bh6t/subitem-1759839952709-wfiuz8f3e"><div class="list-item-paragraph">the last integer in the $n$$^{\mathrm{th}}$ set.<span class="inline-marks">[2]</span></div></li></ol></li><li data-list-item-id="item-1759839965653-x14domfvg" data-list-item-path="item-1759839965653-x14domfvg"><div class="list-item-paragraph">Show that the sum of the terms in the $n$$^{\mathrm{th}}$ set is given by ${{3}^{n}}\left[ A\left( {{3}^{n-2}} \right)+B \right]$, where&nbsp;$A$ and $B$ are integers to be determined.<span class="inline-marks">[4]</span></div></li></ol>

An arithmetic progression $\left\{ {{u}_{n}} \right\}$ is grouped into sets containing $1$, $3$, $9$, … shown below, so that the number of integers in the $n$$^{\mathrm{th}}$ set is ${{3}^{n-1}}$.

$\left\{ 1 \right\}$, $\left\{ 5,\mathrm{ }9,\mathrm{ }13 \right\}$, $\left\{ 17,\mathrm{ }21,\mathrm{ }25,\mathrm{ }29,\mathrm{ }33,\mathrm{ }37,\mathrm{ }41,\mathrm{ }45,\mathrm{ }49 \right\}$, …

2007 HCI P1 Q9

<ol class="i-list"><li><div class="list-item-paragraph">The arithmetic progression is grouped into sets of integers, such that the ${{n}^{\mathrm{th}}}$ set contains $n$ integers as shown.</div><div class="list-item-paragraph" style="text-align: center">$\left\{ 1 \right\},\left\{ 3,5 \right\},\left\{ 7,9,11 \right\},\left\{ 13,15,17,19 \right\},...\,\,...$</div><ol class="abc-list"><li><div class="list-item-paragraph">Find the total number of terms in the first $n$ sets, and hence show that the last term of the ${{n}^{\mathrm{th}}}$ set is ${{n}^{2}}+n-1$.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Find the first term of the ${{n}^{\mathrm{th}}}$ set.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Show that the sum of the terms in the ${{n}^{\mathrm{th}}}$ set is ${{n}^{3}}$.</div><p style="text-align: right;">[1]</p></li></ol></li><li><div class="list-item-paragraph">Hence, prove that ${{1}^{3}}+{{2}^{3}}+{{3}^{3}}+......+{{n}^{3}}=\frac{1}{4}{{n}^{2}}{{\left( n+1 \right)}^{2}}$.</div><p style="text-align: right;">[2]</p></li></ol>

Summing Polynomial Series

2021 CJC Promo Q8

<p>An arithmetic progression has first term $1$ and common difference $2$. The terms of this arithmetic progression are grouped into sets containing $1$, $2$, $3$, $4$, … terms, as indicated below.</p>
<p>$\{1\}$, $\{3, 5\}$, $\{7, 9, 11\}$, $\{13, 15, 17, 19\}$, $\{21, 23, 25, 27, 29\}$, …</p>
<p>Find the last term in the $100$$^{\mathrm{th}}$ set.</p><p style="text-align: right;">[3]</p>

An arithmetic progression has first term $1$ and common difference $2$. The terms of this arithmetic progression are grouped into sets containing $1$, $2$, $3$, $4$, … terms, as indicated below.

$\{1\}$, $\{3, 5\}$, $\{7, 9, 11\}$, $\{13, 15, 17, 19\}$, $\{21, 23, 25, 27, 29\}$, …

Find the last term in the $100$$^{\mathrm{th}}$ set.

Math Question Bank