A Math

A Math (IGCSE)

E Math

E Math (IGCSE)

H1 Math

H2 Math

IB AA HL Math

IB AA SL Math

IB AI HL Math

IB AI SL Math

IP Math

N(A) A Math

N(A) E Math

PSLE

PSLE Foundation Mathematics

<p>The sum, $S_n$, of the first $n$ terms of a sequence $u_1,u_2,u_3,\ldots$ is given by \[S_n=\frac16+\frac1{n+3}-\frac2{n+4}.\]</p>
<ol class="abc-list"><li data-list-item-id="nyjc-q05-item01" data-list-item-path="nyjc-q05-item01"><div class="list-item-paragraph">Give a reason why the series $\displaystyle \sum_{r=1}^{\infty}u_r$ converges and write down the value of the sum to infinity.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="nyjc-q05-item02" data-list-item-path="nyjc-q05-item02"><div class="list-item-paragraph">Show that&nbsp;$\displaystyle u_n=\frac{n}{(n+2)(n+3)(n+4)}$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="nyjc-q05-item03" data-list-item-path="nyjc-q05-item03"><div class="list-item-paragraph">It is given that&nbsp;$\displaystyle 5\sum_{r=k+1}^{\infty}u_r=9\sum_{r=1}^{k}u_r$. Find the value of $k$.<span class="inline-marks">[3]</span></div></li></ol>

The sum, $S_n$, of the first $n$ terms of a sequence $u_1,u_2,u_3,\ldots$ is given by \[S_n=\frac16+\frac1{n+3}-\frac2{n+4}.\]

Give a reason why the series $\displaystyle \sum_{r=1}^{\infty}u_r$ converges and write down the value of the sum to infinity.

Show that $\displaystyle u_n=\frac{n}{(n+2)(n+3)(n+4)}$.

It is given that $\displaystyle 5\sum_{r=k+1}^{\infty}u_r=9\sum_{r=1}^{k}u_r$. Find the value of $k$.

Manipulating Summation Limits

Application of Sum to Infinity

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

2026 NYJC P1 Q5

Sequence and Series

<ol class="abc-list"><li data-list-item-id="item-a2" data-list-item-path="item-a2"><div class="list-item-paragraph">The sum, $S_n$, of the first $n$ terms of a sequence $u_1, u_2, u_3, \dots$ is given by $S_n = 75 - \frac{3^{n+1}}{5^{n-2}}$.<br>Show that&nbsp;$u_n = k\left(\frac{3}{5}\right)^{n-1}$, where&nbsp;$k$ is a constant to be determined. Find $S_\infty$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-b2" data-list-item-path="item-b2"><div class="list-item-paragraph">Another sequence of numbers $v_1, v_2, v_3, \dots$ is such that&nbsp;$\sum_{i=1}^{n} v_i = \frac{2n}{2n + 1}$.<br>Find the exact value of $\sum_{i=11}^{\infty} v_i$.<span class="inline-marks">[3]</span></div></li></ol>

Sum to infinity

2024 VJC Promo Q4

<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">In a geometric progression, the first term is 12 and the fourth term is $-\frac{3}{2}$. The sum of the first $n$ terms of the progression is given by $S_n$ and the sum to infinity of the progression is given by $S$. Find the least value of $n$ for which $\left| S - S_n \right| &lt; 0.001$.<span class="inline-marks">[5]</span></div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">An arithmetic progression has the first term 1000 and common difference $-1.5$. Calculate the value of the first negative term of the progression and the sum of all the positive terms.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">The sum of the first $n$ terms of a series is given by $S_n = \frac{1}{3}n(n + 12)$. By finding an expression for the $n$$^{th}$ term of the series, show that this is an arithmetic series. Hence, state the common difference of the arithmetic series.<span class="inline-marks">[4]</span></div></li></ol>

Problems involving inequalities

Proving the progression

Arithmetic Progression

2024 MI PRU2 P1 Q12

<p>The sum, $S_n$, of the first $n$ terms of the sequence $u_1, u_2, u_3, \dots$ is given by</p>
<p style="text-align: center">\[S_n = n^2 + kn,\]</p>
<p>where&nbsp;$k$ is a non-zero constant.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Find $u_n$ in terms of $n$ and $k$.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Hence determine if the sequence is an arithmetic progression.<span class="inline-marks">[2]</span></div></li></ol>

The sum, $S_n$, of the first $n$ terms of the sequence $u_1, u_2, u_3, \dots$ is given by

2024 MI PRU1 P1 Q1

<ol class="abc-list"><li><div class="list-item-paragraph">In a geometric progression, the first term is $12$ and the fourth term is $-\frac{3}{2}$. The sum of the first $n$ terms of the 	progression is given by ${{S}_{n}}$ and the sum to infinity of the progression is given by $S$. Find the least value of $n$ for 	which $\left| S-{{S}_{n}} \right|&lt;0.001$. 	<span class="inline-marks">[5]</span></div></li><li><div class="list-item-paragraph">An arithmetic progression has the first term $1000$ and common difference $-1.5$. Calculate the value of the first 	negative term of the progression and the sum of all the positive terms. 	<span class="inline-marks">[4]</span></div></li><li><div class="list-item-paragraph">The sum of the first $n$ terms of a series is given by ${{S}_{n}}=\frac{1}{3}n\left( n+12 \right)$. By finding an expression for the ${{n}^{\text{th}}}$ term of the 	series, show that this is an arithmetic series. Hence, state the common difference of the arithmetic series.  <span class="inline-marks">[4]</span></div></li></ol>

<p>The sum of the first $n$ terms of the series $T$ is given by $2n^3 - 8n^2 - 4n$.</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Find an expression for&nbsp;$t_n$, the $n$th term of series $T$.<span class="inline-marks">[2]</span></div></li></ol>
<p>The $n$th term of the series $U$ is given by $u_n = 50n - 204$.</p>
<ol class="abc-list" start="2"><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">Find the values of $n$ for which $u_n = t_n$.<span class="inline-marks">[2]</span></div></li></ol>
<p>The $n$th term of the series $V$ is given by $v_n = 3n + 16$.</p>
<ol class="abc-list" start="3"><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">Find the smallest number greater than $100$ that is in both series $U$ and series $V$.<span class="inline-marks">[2]</span></div></li></ol>
<p>The $n$th term of the series $W$ is given by $w_n = 3n^2 - 5n + 7$.</p>
<ol class="abc-list" start="4"><li data-list-item-id="item-d" data-list-item-path="item-d"><ol class="i-list"><li data-list-item-id="item-d-i" data-list-item-path="item-d/item-d-i"><div class="list-item-paragraph">Explain why all the terms in series $W$ are odd numbers.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-d-ii" data-list-item-path="item-d/item-d-ii"><div class="list-item-paragraph">Hence explain why series $U$ and series $W$ do not have any terms in common.<span class="inline-marks">[1]</span></div></li></ol></li></ol>

The sum of the first $n$ terms of the series $T$ is given by $2n^3 - 8n^2 - 4n$.

Find an expression for $t_n$, the $n$th term of series $T$.

The $n$th term of the series $U$ is given by $u_n = 50n - 204$.

Find the values of $n$ for which $u_n = t_n$.

The $n$th term of the series $V$ is given by $v_n = 3n + 16$.

Find the smallest number greater than $100$ that is in both series $U$ and series $V$.

The $n$th term of the series $W$ is given by $w_n = 3n^2 - 5n + 7$.

Inter-sequence Relationships

N2024 P2 Q4

<p>The sum, $S_n$, of the first $n$ terms of the sequence $\{u_n\}$ is given by</p>
<p style="text-align: center">\[S_n = 2n^2 - 3n, n = 1, 2, 3, \dots\]</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Find the sum of the terms from the $10$th term to the $23$rd term of $\{u_n\}$.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find $u_n$ in terms of $n$.</span></div></li></ol>

The sum, $S_n$, of the first $n$ terms of the sequence $\{u_n\}$ is given by

Find the sum of the terms from the $10$th term to the $23$rd term of $\{u_n\}$.

Recovering term from sequence example

<p>The $n$ th term of a series $G$ is given by $g_n = 3n^2 - 17n + 17$, where&nbsp;$n \geq 1$.</p>
<ol class="abc-list"><li data-list-item-id="item-a-2" data-list-item-path="item-a-2"><div class="list-item-paragraph">Given that&nbsp;$\sum_{r=1}^{n} r^2 = \frac{n}{6}(n+1)(2n+1)$, find $\sum_{r=m+1}^{n} g_r$. (You need not simplify your answer.)<span class="inline-marks">[4]</span></div></li></ol>
<p>The $n$th term of a series $H$ is given by $h_n = 7(5^{n-3}) + 10$, where&nbsp;$n \geq 1$.</p>
<ol class="abc-list" start="2"><li data-list-item-id="item-b-2" data-list-item-path="item-b-2"><div class="list-item-paragraph">State the smallest number that can be found in both series $H$ and $G$.<span class="inline-marks">[1]</span></div></li></ol>
<p>The sum of the first $n$ terms of a series $J$ is given by $\frac{b}{2}(3^n - 1)$, where&nbsp;$b$ is a positive odd integer.</p>
<ol class="abc-list" start="3"><li data-list-item-id="item-c-2" data-list-item-path="item-c-2"><div class="list-item-paragraph">By finding the $n$th term of series $J$, explain why each of the terms in series $J$ is a term of the arithmetic progression with first term $1$ and common difference $2$.<span class="inline-marks">[3]</span></div></li></ol>

The $n$ th term of a series $G$ is given by $g_n = 3n^2 - 17n + 17$, where $n \geq 1$.

Given that $\sum_{r=1}^{n} r^2 = \frac{n}{6}(n+1)(2n+1)$, find $\sum_{r=m+1}^{n} g_r$. (You need not simplify your answer.)

The $n$th term of a series $H$ is given by $h_n = 7(5^{n-3}) + 10$, where $n \geq 1$.

State the smallest number that can be found in both series $H$ and $G$.

The sum of the first $n$ terms of a series $J$ is given by $\frac{b}{2}(3^n - 1)$, where $b$ is a positive odd integer.

By finding the $n$th term of series $J$, explain why each of the terms in series $J$ is a term of the arithmetic progression with first term $1$ and common difference $2$.

Summing Polynomial Series

2025 NYJC P1 Q5

<p>The sum of the first $n$ terms of a sequence, $u_r$, is given by \[\sum_{r=1}^{n} u_r = 1 - \frac{n}{(n+1)!}.\]</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find $u_n$ in terms of $n$, expressing your answer as a single algebraic fraction.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Explain why $\displaystyle \sum_{r=5}^{n} u_r &lt; \frac{1}{30}$, for all $n \ge 5$.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Explain why $\displaystyle \sum_{r=1}^{\infty} u_r$ is a convergent series.<span class="inline-marks">[1]</span></div></li></ol>

The sum of the first $n$ terms of a sequence, $u_r$, is given by \[\sum_{r=1}^{n} u_r = 1 - \frac{n}{(n+1)!}.\]

Find $u_n$ in terms of $n$, expressing your answer as a single algebraic fraction.

<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">The ninth, fifth and second terms of an arithmetic progression are successive terms of a geometric progression with first term $a$ and common ratio $r$, where&nbsp;$r \neq 1$ and $a &gt; 0$.</div><ol class="i-list"><li data-list-item-id="item-a-i" data-list-item-path="item-a/item-a-i"><div class="list-item-paragraph">Find the value of $r$ and deduce that the geometric series is convergent.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-a-ii" data-list-item-path="item-a/item-a-ii"><div class="list-item-paragraph"><p>Using the value of $r$ found in<strong> (i)</strong>, find the least value of $n$ such that the sum of all the terms after the $n^{\mathrm{th}}$ term of the geometric progression is less than $1$% of its sum of the first $n$ terms.</p><span class="inline-marks">[3]</span></div></li></ol></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">The sum, $S_n$, of the first $n$ terms of another sequence is given by $S_n = n \ln 2^{q(n+1)}$, where&nbsp;$q$ is a constant. Prove that the sequence follows an arithmetic progression.<span class="inline-marks">[4]</span></div></li></ol>

The ninth, fifth and second terms of an arithmetic progression are successive terms of a geometric progression with first term $a$ and common ratio $r$, where $r \neq 1$ and $a > 0$.

The sum, $S_n$, of the first $n$ terms of another sequence is given by $S_n = n \ln 2^{q(n+1)}$, where $q$ is a constant. Prove that the sequence follows an arithmetic progression.

2025 ASRJC P1 Q8

<p>The sum of the first $n$ terms of a sequence is given by $S_n = 4 - \frac{(-1)^n}{4^{n-1}}$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Find an expression for the $n$ th term of the sequence and show that the sequence is a geometric progression.</div></li><li><div class="list-item-content"><span class="list-item-text">Find the sum to infinity, $S$, of the progression and the least value of $n$ for which the magnitude of the difference between $S_n$ and $S$ is less than $0.001$.</span></div></li></ol>

The sum of the first $n$ terms of a sequence is given by $S_n = 4 - \frac{(-1)^n}{4^{n-1}}$.

Find the sum to infinity, $S$, of the progression and the least value of $n$ for which the magnitude of the difference between $S_n$ and $S$ is less than $0.001$.

2007 NJC Promo Q7

<p>The sum of the first $n$ terms of a sequence is given by ${{S}_{n}}=2{{n}^{2}}+n$. Prove that the series in an arithmetic series and state the common difference.</p>

The sum of the first $n$ terms of a sequence is given by ${{S}_{n}}=2{{n}^{2}}+n$. Prove that the series in an arithmetic series and state the common difference.

Worked Example 1

<p>The sum of the first $n$ terms of a sequence is given by $S_{n} = 5^{n} - 1$. State the first term and prove that the series is a geometric series. Thus, state the common ratio.</p>

The sum of the first $n$ terms of a sequence is given by $S_{n} = 5^{n} - 1$. State the first term and prove that the series is a geometric series. Thus, state the common ratio.

<p>The sum of the first $n$ terms of a sequence is given by $S_{n} = 6 - \dfrac{3^{n+1}}{2^{n-1}}$. State the first term and prove that the series is a geometric series. Thus, state the common ratio.</p>

The sum of the first $n$ terms of a sequence is given by $S_{n} = 6 - \dfrac{3^{n+1}}{2^{n-1}}$. State the first term and prove that the series is a geometric series. Thus, state the common ratio.

<p>The sum of the first $n$ terms of a sequence is given by $S_{n} = 3n - n^{2}$. Prove that the series is an arithmetic series and state the common difference.</p>

The sum of the first $n$ terms of a sequence is given by $S_{n} = 3n - n^{2}$. Prove that the series is an arithmetic series and state the common difference.

<p>The sum of the first $n$ terms of a sequence is given by $S_{n} = 2n^{2} + n$. Prove that the series is an arithmetic series and state the common difference.</p>

The sum of the first $n$ terms of a sequence is given by $S_{n} = 2n^{2} + n$. Prove that the series is an arithmetic series and state the common difference.

<ol class="abc-list"><li><div class="list-item-paragraph">The sum, ${{S}_{n}}$, of the first $n$ terms of a series is given by ${{S}_{n}}={{n}^{2}}+2n$. Show that this is an arithmetic series. Find the values of the first term and the common difference.<span class="inline-marks">[4]</span></div></li><li><div class="list-item-paragraph">In a geometric progression, the first term is $12$ and the sixth term is $-\frac{3}{8}$. Let the sum of the first $n$ terms of the progression and the sum to infinity be ${{S}_{n}}$ and $S$ respectively. Find the least value of $n$ for which the difference between ${{S}_{n}}$ and $S$ is less than $0.001$.)<span class="inline-marks">[4]</span></div></li></ol>

2024 NYJC Promo Q6

<p>The sum, ${{S}_{n}}$, of the first $n$ terms of the sequence ${{u}_{1}}$, ${{u}_{2}}$, ${{u}_{3}}$, $\ldots $ is given by</p>
<p style="text-align: center">${{S}_{n}}={{n}^{2}}+kn$,</p>
<p>where&nbsp;$k$ is a non-zero constant.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Find ${{u}_{n}}$ in terms of $n$ and $k$.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Hence determine if the sequence is an arithmetic progression.<span class="inline-marks">[2]</span></div></li></ol>

The sum, ${{S}_{n}}$, of the first $n$ terms of the sequence ${{u}_{1}}$, ${{u}_{2}}$, ${{u}_{3}}$, $\ldots $ is given by

<ol class="abc-list"><li><div class="list-item-paragraph">The sum, ${{S}_{n}}$, of the first $n$ terms of an arithmetic progression is given by ${{S}_{n}}=pn+q{{n}^{2}}$.</div><div class="list-item-paragraph">Given also that ${{S}_{3}}=6$ and ${{S}_{5}}=11$,</div><ol class="i-list"><li><div class="list-item-paragraph">Find the values of $p$ and $q$.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Deduce, or find otherwise, an expression for the $n$-th term and the value of the common difference.</div><p style="text-align: right;">[3]</p></li></ol></li><li><div class="list-item-paragraph">Find the set of values of $\theta $ lying in the interval $-\frac{1}{2}\pi <\theta <\frac{1}{2}\pi $ such that the sum to infinity of the geometric series $1+\sin \theta +{{\sin }^{2}}\theta +\ldots $ is greater than $2$.</div><p style="text-align: right;">[4]</p></li></ol>

N1984 P1 Q1

<p>The sum, ${{S}_{n}}$, of the first $n$ terms of a sequence ${{u}_{1}}$, ${{u}_{2}}$,${{u}_{3}}$, … is given by ${{S}_{n}}={{n}^{2}}\left( An+B \right)$, where $A$ and $B$ are non-zero constants.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Find an expression for ${{u}_{n}}$ in terms of $A$, $B$ and $n$, simplifying your answer.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">It is also given that the second term is $6$ and the sixth term is $218$. Find the values of $A$ and $B$.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Determine if the sum to infinity exists.</div><p style="text-align: right;">[1]</p></li></ol>

The sum, ${{S}_{n}}$, of the first $n$ terms of a sequence ${{u}_{1}}$, ${{u}_{2}}$,${{u}_{3}}$, … is given by ${{S}_{n}}={{n}^{2}}\left( An+B \right)$, where $A$ and $B$ are non-zero constants.

Math Question Bank