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<p>It is given that&nbsp;$ \sum_{r=1}^{n} r^3 = \frac{n^2(n+1)^2}{4} $.</p>
<ol class="abc-list"><li data-list-item-id="h2-2025-vjc-promo-q3-vjc-q3-item-block-1-item-0" data-list-item-path="h2-2025-vjc-promo-q3-vjc-q3-item-block-1-item-0"><div class="list-item-paragraph">Find $ \sum_{r=4}^{N+3} (r-2)^3 $ in terms of $N$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="h2-2025-vjc-promo-q3-vjc-q3-item-block-1-item-1" data-list-item-path="h2-2025-vjc-promo-q3-vjc-q3-item-block-1-item-1"><div class="list-item-paragraph">Hence, find the largest value of $N$ for which $ \sum_{r=4}^{N+3} \left[(2r-4)^3+2^r\right] &lt; 10^5 $.<span class="inline-marks">[3]</span></div></li></ol>

It is given that $ \sum_{r=1}^{n} r^3 = \frac{n^2(n+1)^2}{4} $.

Find $ \sum_{r=4}^{N+3} (r-2)^3 $ in terms of $N$.

Hence, find the largest value of $N$ for which $ \sum_{r=4}^{N+3} \left[(2r-4)^3+2^r\right] < 10^5 $.

Sigma Notation By Substitution

Summations Involving APGP

2025 VJC Promo Q3

Sequence and Series

<p>Find</p>
<ol class="i-list"><li><div class="list-item-paragraph">$\sum\limits_{r=4}^{n+1}{(\ln {{3}^{r}}}+{{5}^{-r}})$</div></li></ol>

Summations Involving APGP Q3

<p>Find</p>
<ol class="i-list"><li><div class="list-item-paragraph">$\sum\limits_{r=3}^{n}{(3nr+n+{{3}^{r}})}$</div></li><li><div class="list-item-paragraph">$\sum\limits_{r=0}^{n}{(nr+n+{{2}^{r}})}$</div></li></ol>

Summations Involving APGP Q2

<ol class="abc-list"><li><div class="list-item-paragraph">A geometric sequence ${{x}_{1}}$, ${{x}_{2}}$, ${{x}_{3}}$, … has first term $a$ and common ratio $r$, where $a>0$, $r>0$. The sequence of numbers ${{y}_{1}}$, ${{y}_{2}}$, ${{y}_{3}}$, … satisfies the relation ${{y}_{n}}={{\log }_{4}}\left( {{x}_{n}} \right)$ for $n\in {{\mathbb{Z}}^{+}}$.</div><ol class="i-list"><li><div class="list-item-paragraph">If the product of ${{x}_{5}}$ and ${{x}_{21}}$ is $4096$, find the value of $\sum\limits_{k=1}^{25}{{{\log }_{4}}{{x}_{k}}}$.</div><p style="text-align: right;">[4]</p></li><li><div class="list-item-paragraph">Show that ${{y}_{1}}$, ${{y}_{2}}$, ${{y}_{3}}$, … is an arithmetic sequence.</div><p style="text-align: right;">[2]</p></li></ol></li><li><div class="list-item-paragraph">The output of an oil field in any year is $6\%$ less than in the preceding year. Prove that the total output of the oil mined from the oil field cannot exceed $17$ times the output in the first year.</div><p style="text-align: right;">[2]</p><div class="list-item-paragraph">It is decided to close the mine when the total output exceeds $16$ times the output in the first year. Suppose the oil field was first mined in the year 1997, determine the earliest year in which the oil field will be closed.</div><p style="text-align: right;">[2]</p></li></ol>

Problems involving inequalities

Proving the progression

Geometric Progression

Sum to infinity

2015 NYJC P1 Q9

<p>A sequence ${{u}_{1}}$, ${{u}_{2}}$, ${{u}_{3}}$ ... is such that ${{u}_{n+1}}=3{{u}_{n}}+Pn$, where $P$ is a constant and $n\in {{\mathbb{Z}}^{+}}$.</p>
<p>The terms of the sequence are defined by their previous terms, for example, ${{u}_{2}}=3{{u}_{1}}+P$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Given that ${{u}_{1}}=1$ and ${{u}_{2}}=7$, find $P$ and ${{u}_{3}}$.</div><p style="text-align: right;">[2]</p></li></ol>
<p>lt is known that the ${{n}^{\mathrm{th}}}$ term of this sequence is given by</p>
<p style="text-align: center">${{u}_{n}}=a\left( {{3}^{n}} \right)+bn+c$,</p>
<p>where $a$, $b$ and $c$ are constants.</p>
<ol class="i-list" start="2"><li><div class="list-item-paragraph">Find $a$, $b$ and $c$.</div><p style="text-align: right;">[4]</p></li><li><div class="list-item-paragraph">Find $\sum\limits_{r=1}^{n}{{{u}_{r}}}$ in terms of $n$.</div><p style="text-align: right;">[4]</p></li></ol>

A sequence ${{u}_{1}}$, ${{u}_{2}}$, ${{u}_{3}}$ ... is such that ${{u}_{n+1}}=3{{u}_{n}}+Pn$, where $P$ is a constant and $n\in {{\mathbb{Z}}^{+}}$.

The terms of the sequence are defined by their previous terms, for example, ${{u}_{2}}=3{{u}_{1}}+P$.

lt is known that the ${{n}^{\mathrm{th}}}$ term of this sequence is given by

${{u}_{n}}=a\left( {{3}^{n}} \right)+bn+c$,

Arithmetic Progression

2020 CJC Promo Q9

<ol class="abc-list"><li data-list-item-id="item-1759735334250-i993i7imo" data-list-item-path="item-1759735334250-i993i7imo"><div class="list-item-paragraph">It is given that&nbsp;$\sum\limits_{r=1}^{n}{{{r}^{2}}=\frac{1}{6}n\left( n+1 \right)\left( 2n+1 \right)}$.</div><ol class="i-list"><li data-list-item-id="subitem-1759735339981-tj6dep77t" data-list-item-path="item-1759735334250-i993i7imo/subitem-1759735339981-tj6dep77t"><div class="list-item-paragraph">Find $\sum\limits_{r=1}^{n}{\left( {{2}^{r+1}}+3r-{{r}^{2}} \right)}$ in the form $A\left( {{2}^{n}}-1 \right)+\mathrm{f}\left( n \right)$, where&nbsp;$A$ is a constant and $\mathrm{f}\left( n \right)$ is in fully factorized form.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="subitem-1759735340813-k8zz65q0m" data-list-item-path="item-1759735334250-i993i7imo/subitem-1759735340813-k8zz65q0m"><div class="list-item-paragraph">Using your answer in part <strong>(a)(i)</strong>, find $\sum\limits_{r=4}^{N}{\left( {{2}^{r}}+3r-{{\left( r-1 \right)}^{2}} \right)}$, leaving your answer in the form $B\left( {{2}^{N}} \right)+C\left[ N\left( N-1 \right)\left( 5-N \right) \right]+DN+E$ where&nbsp;$B$, $C$, $D$ and $E$ are constants to be determined.<span class="inline-marks">[4]</span></div></li></ol></li></ol>

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