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PSLE

PSLE Foundation Mathematics

<p>The sum of the first 5 terms of an arithmetic progression, where the common difference is non-zero, is 105. It is further given that the first, third and fourth terms of the arithmetic progression are the first 3 terms of a geometric progression. Show that the common difference of the arithmetic progression is $-\frac{21}{2}$.<span class="inline-marks">[3]</span></p>
<p>Deduce that the geometric progression is convergent.<span class="inline-marks">[2]</span></p>
<p>Let&nbsp;$S$ be the sum to infinity of the geometric progression, and $A_n$ be the sum to $n$ terms of the arithmetic progression. Find the least value of $n$ such that the sum of $7S$ and $A_n$ is not positive.<span class="inline-marks">[4]</span></p>

The sum of the first 5 terms of an arithmetic progression, where the common difference is non-zero, is 105. It is further given that the first, third and fourth terms of the arithmetic progression are the first 3 terms of a geometric progression. Show that the common difference of the arithmetic progression is $-\frac{21}{2}$.

Deduce that the geometric progression is convergent.

Let $S$ be the sum to infinity of the geometric progression, and $A_n$ be the sum to $n$ terms of the arithmetic progression. Find the least value of $n$ such that the sum of $7S$ and $A_n$ is not positive.

Problems involving inequalities

Sum to infinity

Arithmetic Progression

Inter-sequence Relationships

VJC P1 Q13

Sequence and Series

<p>An arithmetic series has first term $a$ and common difference $d$, where&nbsp;$a$ is non-zero. The $8$th, $4$th and $2$nd term of the arithmetic series are the first three consecutive terms of a geometric series respectively. It is also given that the terms of the arithmetic series are increasing.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Show that&nbsp;$a = d$.<span class="inline-marks">[4]</span></div></li></ol>
<p>It is given that the sum of the first $20$ terms of the arithmetic series is $630$.</p>
<ol class="i-list" start="2"><li><div class="list-item-paragraph">Find the value of $a$.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Hence find the exact sum of the first $10$ terms of the geometric series.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Give a reason why the geometric series converges and find the sum to infinity of this series.<span class="inline-marks">[2]</span></div></li></ol>

An arithmetic series has first term $a$ and common difference $d$, where $a$ is non-zero. The $8$th, $4$th and $2$nd term of the arithmetic series are the first three consecutive terms of a geometric series respectively. It is also given that the terms of the arithmetic series are increasing.

It is given that the sum of the first $20$ terms of the arithmetic series is $630$.

Geometric Progression

2024 MI PRU1 Q8

<p>The annual wage of a certain occupation offered by Company $P$ begins with an initial amount of $\$9500$ and increases by $\$400$ every year for&nbsp;$14$ years till it reaches $\$15100$ and remains constant at $\$15100$ thereafter.</p>
<p>Let&nbsp;$p(n) = a_1 + a_2 + a_3 + ... + a_n$, where&nbsp;$a_i$ denotes the annual wage in the $i^{\mathrm{th}}$ year offered by Company $P$ for&nbsp;$i = 1, 2, 3, \dots, n$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Show that&nbsp;$p(n) = \begin{cases} 200n^2 + 9300n & \text{for}\hspace{0.5em} n \le 15, \\ 15100n + A & \text{for}\hspace{0.5em} n &gt; 15, \end{cases}$ </div><div class="list-item-paragraph">where&nbsp;$A$ is a constant to be determined.<span class="inline-marks">[4]</span></div></li><li><div class="list-item-paragraph">The annual wage of the same occupation offered by Company $Q$ begins with the same initial amount of $\$9500$ but it increases at a fixed rate of $r \, \%$ every year.</div><div class="list-item-paragraph">Sam has been offered a job for the same occupation at both Company $P$ and $Q$. Find the value of $r$, correct to $2$ decimal places, such that the total annual wages earned by Sam if he works for Company $P$ for&nbsp;$15$ years is equivalent to the total annual wages earned if he works for Company $Q$ for&nbsp;$12$ years.<span class="inline-marks">[4]</span></div></li></ol>

The annual wage of a certain occupation offered by Company $P$ begins with an initial amount of $\$9500$ and increases by $\$400$ every year for $14$ years till it reaches $\$15100$ and remains constant at $\$15100$ thereafter.

Let $p(n) = a_1 + a_2 + a_3 + ... + a_n$, where $a_i$ denotes the annual wage in the $i^{\mathrm{th}}$ year offered by Company $P$ for $i = 1, 2, 3, \dots, n$.

2014 RI Promo Q6

<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 $w_n = 3n^2 - 5n + 7$.

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

N2024 P2 Q4

<p>An arithmetic series has positive first term $a$ and common difference $d$. The first, fifth and second terms of the arithmetic series are the first three consecutive terms of a geometric series respectively. It is also given that the terms of the arithmetic series are decreasing.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Show that&nbsp;$d = -\frac{7}{16}a$.</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Give a reason why the geometric series converges.</span><span class="inline-marks">[2]</span></div></li></ol>
<p>It is given that the sum to infinity of the geometric series is $\frac{32}{7}$.</p>
<ol class="abc-list" start="3"><li><div class="list-item-content"><span class="list-item-text">Find the sum of the first twenty terms of the arithmetic series.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Given that the sum of all the terms after the $n$th term of the geometric series is greater than $2$% of the sum of the first $n$ terms of the geometric series, find the possible values of $n$.</span><span class="inline-marks">[4]</span></div></li></ol>

An arithmetic series has positive first term $a$ and common difference $d$. The first, fifth and second terms of the arithmetic series are the first three consecutive terms of a geometric series respectively. It is also given that the terms of the arithmetic series are decreasing.

Give a reason why the geometric series converges.

It is given that the sum to infinity of the geometric series is $\frac{32}{7}$.

Find the sum of the first twenty terms of the arithmetic series.

Given that the sum of all the terms after the $n$th term of the geometric series is greater than 2% of the sum of the first $n$ terms of the geometric series, find the possible values of $n$.

2025 RI Promo Q11

<p>An arithmetic series has first term $a$ and common difference $d$, where&nbsp;$a$ and $d$ are non-zero real constants. A convergent geometric series has first term $b$ and common ratio $r$, where&nbsp;$b &gt; 0$ and $r$ is a non-zero real constant. The sum of the first $n$ terms of the geometric series is denoted by $S$. It is given that the $7$th, $10$th and $11$th terms of the arithmetic series are equal to the $3$rd, $6$th and $10$th terms of the geometric series respectively.</p>
<ol class="abc-list"><li data-list-item-id="item-3a" data-list-item-path="item-3a"><div class="list-item-paragraph">Show that&nbsp;$r$ satisfies the equation $3r^7 - 4r^3 + 1 = 0$. Solve this equation, giving your answer correct to $4$ decimal places.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-3b" data-list-item-path="item-3b"><div class="list-item-paragraph">A new series is formed by taking the even numbered terms of the geometric series. Find the smallest value of $n$ for which $S$ differs from the sum to infinity of the new series by at least $1.78b$.<span class="inline-marks">[4]</span></div></li></ol>

An arithmetic series has first term $a$ and common difference $d$, where $a$ and $d$ are non-zero real constants. A convergent geometric series has first term $b$ and common ratio $r$, where $b > 0$ and $r$ is a non-zero real constant. The sum of the first $n$ terms of the geometric series is denoted by $S$. It is given that the $7$th, $10$th and $11$th terms of the arithmetic series are equal to the $3$rd, $6$th and $10$th terms of the geometric series respectively.

Show that $r$ satisfies the equation $3r^7 - 4r^3 + 1 = 0$. Solve this equation, giving your answer correct to $4$ decimal places.

A new series is formed by taking the even numbered terms of the geometric series. Find the smallest value of $n$ for which $S$ differs from the sum to infinity of the new series by at least $1.78b$.

Subsequences of Progressions

2025 TMJC P1 Q4

<p>Mabel and Janice decided to start a $5$-year savings plan beginning in January $2026$.</p>
<p>Mabel saves using a piggy bank. At the start of January $2026$, she deposits $\$101$. Each subsequent month, she increases her deposit by $\$1$—so she deposits $$101$ in January, $\$102$ in February, $\$103$ in March, and so on, until $\$112$ in December. At the start of each new year, she resets her monthly deposit to $\$101$ in January and repeats the same pattern through December. She continues this routine from $2026$ to $2030$, inclusive.</p>
<p>Janice, on the other hand, deposits $$100$ at the start of every month into a bank account that earns $0.3$% interest per month, with interest calculated and added into the account at the end of each month.</p>
<ol class="abc-list"><li data-list-item-id="item-6-a" data-list-item-path="item-6-a"><div class="list-item-paragraph">Show that Janice will have more money in her savings account than Mabel has in her piggy bank at the end of December $2030$.<span class="inline-marks">[5]</span></div></li><li data-list-item-id="item-6-b" data-list-item-path="item-6-b"><div class="list-item-paragraph">Find the month and year when Janice’s savings first exceed Mabel’s savings.<span class="inline-marks">[4]</span></div></li></ol>

Mabel and Janice decided to start a $5$-year savings plan beginning in January $2026$.

Mabel saves using a piggy bank. At the start of January $2026$, she deposits $$101$. Each subsequent month, she increases her deposit by $$1$—so she deposits $$101$ in January, $$102$ in February, $$103$ in March, and so on, until $$112$ in December. At the start of each new year, she resets her monthly deposit to $$101$ in January and repeats the same pattern through December. She continues this routine from $2026$ to $2030$, inclusive.

Janice, on the other hand, deposits $$100$ at the start of every month into a bank account that earns $0.3$% interest per month, with interest calculated and added into the account at the end of each month.

Show that Janice will have more money in her savings account than Mabel has in her piggy bank at the end of December $2030$.

Find the month and year when Janice’s savings first exceed Mabel’s savings.

Annuities and Amortization

2025 TJC P1 Q7

<p>The sequence $\{u_n\}$ is a geometric sequence with first term $a$ and common ratio $\frac{3}{2}$, where&nbsp;$a &gt; 0$.</p>
<p>Another sequence $\{v_n\}$ is defined by $v_n = \frac{9}{u_n}$ for all positive integers $n$.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Show that&nbsp;$\{v_n\}$ is also a geometric sequence.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Given that&nbsp;$v_2 = u_2$, find the value of $a$.</span><span class="inline-marks">[2]</span></div></li></ol>

The sequence $\{u_n\}$ is a geometric sequence with first term $a$ and common ratio $\frac{3}{2}$, where $a > 0$.

Another sequence $\{v_n\}$ is defined by $v_n = \frac{9}{u_n}$ for all positive integers $n$.

Show that $\{v_n\}$ is also a geometric sequence.

Given that $v_2 = u_2$, find the value of $a$.

Proving the progression

2025 TJC P1 Q1

<p>An arithmetic progression has first term $a$ and common difference $d$, where&nbsp;$a$ and $d$ are non-zero. The $3^{\mathrm{rd}}$, $9^{\mathrm{th}}$ and $11^{\mathrm{th}}$ terms of the arithmetic progression are three distinct consecutive terms of a geometric progression.</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Find $d$ in terms of $a$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph"><p>Let the sum of the first $n$ terms of the arithmetic progression be $S_n$, where&nbsp;$S_{1} = 143$. Find the largest possible value of $S_n$.</p><span class="inline-marks">[3]</span></div></li></ol>
<p>The first term of the geometric progression is $b$, where&nbsp;$b$ is positive.</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">Determine whether the geometric series is convergent.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-d" data-list-item-path="item-d"><div class="list-item-paragraph">Show that the sum of all terms after the $n^{\mathrm{th}}$ term of the geometric progression is at most $\frac{1}{2}b$, where&nbsp;$n$ is a positive integer.<span class="inline-marks">[2]</span></div></li></ol>

An arithmetic progression has first term $a$ and common difference $d$, where $a$ and $d$ are non-zero. The $3^{\mathrm{rd}}$, $9^{\mathrm{th}}$ and $11^{\mathrm{th}}$ terms of the arithmetic progression are three distinct consecutive terms of a geometric progression.

The first term of the geometric progression is $b$, where $b$ is positive.

Determine whether the geometric series is convergent.

Show that the sum of all terms after the $n^{\mathrm{th}}$ term of the geometric progression is at most $\frac{1}{2}b$, where $n$ is a positive integer.

2025 SAJC P1 Q5

<p>A sequence is such that&nbsp;$u_1 = N$ where&nbsp;$N$ is a constant, and $u_{n+1} = 3u_n - 8$, for&nbsp;$n \ge 1$.</p>
<ol class="abc-list"><li data-list-item-id="item-seq-a" data-list-item-path="item-seq-a"><div class="list-item-paragraph">Describe how the sequence behaves when</div><ol class="i-list"><li data-list-item-id="item-seq-a-i" data-list-item-path="item-seq-a/item-seq-a-i"><div class="list-item-paragraph">$N = 4$,<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-seq-a-ii" data-list-item-path="item-seq-a/item-seq-a-ii"><div class="list-item-paragraph">$N = 3$.<span class="inline-marks">[1]</span></div></li></ol></li><li data-list-item-id="item-seq-b" data-list-item-path="item-seq-b"><div class="list-item-paragraph">Find the value of $N$ for which $u_5 = 409$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-seq-c" data-list-item-path="item-seq-c"><div class="list-item-paragraph">The sequence $v_n = (-1)^n (u_n + k)$ for&nbsp;$n \ge 1$ where&nbsp;$k$ is a constant, is convergent when&nbsp;$u_1 = 4$. State the value of $k$.<span class="inline-marks">[1]</span></div></li></ol>

A sequence is such that $u_1 = N$ where $N$ is a constant, and $u_{n+1} = 3u_n - 8$, for $n \ge 1$.

Find the value of $N$ for which $u_5 = 409$.

The sequence $v_n = (-1)^n (u_n + k)$ for $n \ge 1$ where $k$ is a constant, is convergent when $u_1 = 4$. State the value of $k$.

Limit of a Sequence and Behaviour of a Sequence

Reverse Iteration of Recurrence Relations

2025 RVHS P2 Q1

<ol class="abc-list"><li data-list-item-id="item-a-2" data-list-item-path="item-a-2"><div class="list-item-paragraph"><p>A geometric series has first term $2$ and common ratio $0.8$. Find algebraically, the least value of $m$ for the sum of the first $4m$ terms of the series to be greater than $99\%$ of the sum to infinity.</p><span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-b-2" data-list-item-path="item-b-2"><div class="list-item-paragraph">A finite arithmetic progression $A$ has&nbsp;$n$ terms, first term $a$ and common difference $d$. In another progression $B$, the $k$th term is obtained by adding the $k$th positive odd integer to the corresponding term of $A$. That is, the first term of $B$ is obtained by adding $1$ to the first term of $A$, the second term of $B$ is obtained by adding $3$ to the second term of $A$ and so on. It is given that the twelfth term of $A$ is $25$, the sum of all the terms of $A$ is $676$ and the sum of all the terms of $B$ is twice the sum of all the terms of $A$.</div><ol class="i-list"><li data-list-item-id="item-b-2-i" data-list-item-path="item-b-2/item-b-2-i"><div class="list-item-paragraph">Find the values of $n$, $a$ and $d$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-b-2-ii" data-list-item-path="item-b-2/item-b-2-ii"><div class="list-item-paragraph">Obtain the sum of the first ten terms of $B$.<span class="inline-marks">[2]</span></div></li></ol></li></ol>

2025 RI P1 Q6

<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 $2$.

Manipulating Summation Limits

Summing Polynomial Series

2025 NYJC P1 Q5

<p>A sequence is defined by $T_{n+2} = T_{n+1} + T_n$ with $T_1 = 3$, $T_2 = 1$ for&nbsp;$n \ge 1$. Another sequence is defined by $r_n = \frac{T_{n+1}}{T_n}$ for&nbsp;$n \ge 1$.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find $T_4$.</span><span class="inline-marks">[1]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Show that&nbsp;$r_{n+1} = 1 + \frac{1}{r_n}$ for all for&nbsp;$n \ge 1$.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">It is given that&nbsp;$r_n$ converges to a limit $L$. Find the exact value of $L$.</span><span class="inline-marks">[3]</span></div></li></ol>
<p>A third sequence $u_n$, for&nbsp;$n \ge 1$, is an arithmetic sequence with first term $5$ and common difference $2$.</p>
<ol class="abc-list" start="4"><li><div class="list-item-paragraph">Find the least value of $n$ such that&nbsp;$\displaystyle \sum_{k=1}^n u_k &gt; 930$.<span class="inline-marks">[2]</span></div></li></ol>

A sequence is defined by $T_{n+2} = T_{n+1} + T_n$ with $T_1 = 3$, $T_2 = 1$ for $n \ge 1$. Another sequence is defined by $r_n = \frac{T_{n+1}}{T_n}$ for $n \ge 1$.

Show that $r_{n+1} = 1 + \frac{1}{r_n}$ for all for $n \ge 1$.

It is given that $r_n$ converges to a limit $L$. Find the exact value of $L$.

A third sequence $u_n$, for $n \ge 1$, is an arithmetic sequence with first term $5$ and common difference $2$.

Generating Terms from Recurrence Relations

2025 MI P2 Q4

<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>A convergent geometric progression has a positive first term $a$ and common ratio $r$ while an arithmetic progression has a non-zero common difference $d$. The second and third term of the geometric progression is the first and fourth term of the arithmetic progression respectively. It is also given that the first term of the geometric progression is equal to the sum of the first $4$ odd-numbered terms in the arithmetic progression.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">By expressing the first term of the arithmetic progression in terms of $a$ and $r$, show that&nbsp;$r^2 = \frac{1}{4}$.</span><span class="inline-marks">[4]</span></div></li><li><div class="list-item-content"><span class="list-item-text">It is given that the sum to infinity of the geometric progression is less than $a$.</span></div><ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find the value of $r$, justifying your answer.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Explain if this implies that for all $n$, the sum of the first $n$ terms of the geometric progression must be less than $a$.</span><span class="inline-marks">[1]</span></div></li></ol></li></ol>

A convergent geometric progression has a positive first term $a$ and common ratio $r$ while an arithmetic progression has a non-zero common difference $d$. The second and third term of the geometric progression is the first and fourth term of the arithmetic progression respectively. It is also given that the first term of the geometric progression is equal to the sum of the first $4$ odd-numbered terms in the arithmetic progression.

By expressing the first term of the arithmetic progression in terms of $a$ and $r$, show that $r^2 = \frac{1}{4}$.

It is given that the sum to infinity of the geometric progression is less than $a$.

2013 NJC P1 Q4

<p>The sum of the first $100$ terms of an arithmetic progression is $10000$; the first, second and fifth terms of this progression are three consecutive terms of a geometric progression. Find the first term, $a$, and the non-zero common difference, $d$, of the arithmetic progression. </p>

The sum of the first $100$ terms of an arithmetic progression is $10000$; the first, second and fifth terms of this progression are three consecutive terms of a geometric progression. Find the first term, $a$, and the non-zero common difference, $d$, of the arithmetic progression. 

<p><strong>Do not use a calculator in answering this question.</strong></p>
<p>An arithmetic sequence ${{u}_{1}}$, ${{u}_{2}}$, ${{u}_{3}}$, $\ldots $ has ${{u}_{1}}=9$ and ${{u}_{3}}=b$, where $b$ is a constant.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Find ${{u}_{2}}$ in terms of $b$.</div><p style="text-align: right;">[2]</p></li></ol>
<p>A geometric sequence ${{v}_{1}}$, ${{v}_{2}}$, ${{v}_{3}}$, $\ldots $ has ${{v}_{1}}=9$ and ${{v}_{3}}=b$.</p>
<ol class="abc-list" start="2"><li><div class="list-item-paragraph">Find the possible values of ${{v}_{2}}$ in terms of $b$.</div><p style="text-align: right;">[2]</p></li></ol>
<p>It is now given that ${{u}_{2}}-{{v}_{2}}=8$, and the geometric sequence ${{v}_{1}}$, ${{v}_{2}}$, ${{v}_{3}}$, $\ldots $ is convergent.</p>
<ol class="abc-list" start="3"><li><div class="list-item-paragraph">Find the value of $b$.</div><div class="list-item-paragraph">Hence find ${{u}_{2}}$ and ${{v}_{2}}$.</div><p style="text-align: right;">[5]</p></li></ol>

**Do not use a calculator in answering this question.**

An arithmetic sequence ${{u}_{1}}$, ${{u}_{2}}$, ${{u}_{3}}$, $\ldots $ has ${{u}_{1}}=9$ and ${{u}_{3}}=b$, where $b$ is a constant.

A geometric sequence ${{v}_{1}}$, ${{v}_{2}}$, ${{v}_{3}}$, $\ldots $ has ${{v}_{1}}=9$ and ${{v}_{3}}=b$.

It is now given that ${{u}_{2}}-{{v}_{2}}=8$, and the geometric sequence ${{v}_{1}}$, ${{v}_{2}}$, ${{v}_{3}}$, $\ldots $ is convergent.

2025 EJC P1 Q7

<p>A geometric sequence ${{u}_{1}}$, ${{u}_{2}}$, ${{u}_{3}}$, … has a common ratio $r$, where $r$ and ${{u}_{1}}$ are positive.</p>
<p>A sequence ${{v}_{1}}$, ${{v}_{2}}$,${{v}_{3}}$, … is given by ${{v}_{n}}=\ln \left( {{u}_{n}} \right)$ for all positive integer $n$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Show that ${{v}_{1}}$, ${{v}_{2}}$,${{v}_{3}}$, … is an arithmetic sequence.</div><p style="text-align: right;">[2]</p></li></ol>
<p>The sequence ${{u}_{1}}$, ${{u}_{2}}$, ${{u}_{3}}$, … is convergent.</p>
<ol class="abc-list" start="2"><li><div class="list-item-paragraph">Find the range of values of the common difference of the sequence ${{v}_{1}}$, ${{v}_{2}}$,${{v}_{3}}$, …</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Given that the sum to infinity of the sequence ${{u}_{1}}$, ${{u}_{2}}$, ${{u}_{3}}$, … is $9$ and ${{v}_{3}}=0$, find the possible values of $r$ and ${{u}_{1}}$.</div><p style="text-align: right;">[4]</p></li></ol>

A geometric sequence ${{u}_{1}}$, ${{u}_{2}}$, ${{u}_{3}}$, … has a common ratio $r$, where $r$ and ${{u}_{1}}$ are positive.

A sequence ${{v}_{1}}$, ${{v}_{2}}$,${{v}_{3}}$, … is given by ${{v}_{n}}=\ln \left( {{u}_{n}} \right)$ for all positive integer $n$.

The sequence ${{u}_{1}}$, ${{u}_{2}}$, ${{u}_{3}}$, … is convergent.

2022 EJC Promo Q8

<p>The fifth, tenth and twentieth term of a convergent geometric progression, ${G}$, are the first three consecutive terms of an arithmetic progression, ${A}$.</p>
<ol class="i-list"><li data-list-item-id="item-1760839937177-psbt8wu4o" data-list-item-path="item-1760839937177-psbt8wu4o"><div class="list-item-paragraph">Determine the common ratio ${G}$.<span class="inline-marks">[3]</span></div></li></ol>
<p>Given that the first term of ${G}$ is ${2}$.</p>
<ol class="i-list" start="2"><li data-list-item-id="item-1760839950954-k9l91lgsg" data-list-item-path="item-1760839950954-k9l91lgsg"><div class="list-item-paragraph">evaluate the sum to infinity for&nbsp;${G}$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1760839952185-xhheuhrbf" data-list-item-path="item-1760839952185-xhheuhrbf"><div class="list-item-paragraph">find the sum of the first ${10}$ odd-numbered terms in ${A}$.<span class="inline-marks">[3]</span></div></li></ol>

The fifth, tenth and twentieth term of a convergent geometric progression, ${G}$, are the first three consecutive terms of an arithmetic progression, ${A}$.

Given that the first term of ${G}$ is ${2}$.

<ol class="abc-list"><li><div class="list-item-paragraph">An arithmetic progression has first term $a$ and common difference $d$, where $a>0$ and $d\ne 0$. The eighth, third and second term of the progression are the first three terms of an infinite geometric progression.</div><ol class="i-list"><li><div class="list-item-paragraph">Find the common ratio of the geometric progression.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Find the exact sum of the odd-numbered terms of the geometric progression in terms of $a$.</div><p style="text-align: right;">[3]</p></li></ol></li><li><div class="list-item-paragraph">A programmer coded a program involving a rabbit-fox chase along a straight path to model the actual hunt for a rabbit by a fox. The rabbit first hop is $1.75$m. In each subsequent hop, the distance covered is $1\%$ less than its previous hop. The fox first leaps $3$ m. In each subsequent leap, the distance covered is $0.02$m less than its previous leap. Initially the rabbit is $60$ m ahead of the fox and assume that the rabbit and the fox start and end each hop and leap at the same time.</div><ol class="i-list"><li><div class="list-item-paragraph">By finding the total distance travelled by the fox and the rabbit after n leaps and hops respectively, find the minimum number of hops and leaps for the fox to catch up with the rabbit.</div><p style="text-align: right;">[4]</p></li><li><div class="list-item-paragraph">Find the number of leaps the fox takes before it comes to a stop. Hence, find the minimum starting distance, in metre, between the fox and the rabbit such that the fox will never catch up with the rabbit. Leave your answer to the nearest integer.</div><p style="text-align: right;">[2]</p></li></ol></li></ol>

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