Series

Determine the coefficients of $x^4$ and $x^5$ in the expansion of $(1 - 2x)^5$.

Feb/March 2016

In an arithmetic progression, the 12th term is $17$, and the total of the first $31$ terms is $1023$.

Feb/March 2016

For the expansion of $(\frac{1}{ax} + 2ax^2)^5$, the coefficient of $x$ is $5$. Determine the value of the constant $a$.

Feb/March 2017

For the expansion of $(1 - px)^5$, the coefficient of $x^3$ is $-2160$. Determine the constant $p$.

Feb/March 2019

The first two terms of a geometric progression are $p$ and $2p$ respectively, where $p$ is a positive constant. The sum of the first $n$ terms is greater than $1000p$. Show that $2^n > 1001$.

Feb/March 2019

In the expansion of $(2x + \frac{a}{x^2})^5$, the coefficient of $\frac{1}{x}$ is $720$.

Feb/March 2020

A woman’s basic salary in her first year at a certain company is $\$30\,000$, and when that year ends she is also given a bonus of $\$600$.

Feb/March 2020

Write down the first three terms, in ascending powers of $x$, of the expansion of $(1 + x)^5$.

Feb/March 2021

The first term in the progression is $\cos \theta$, with $0 < \theta < \tfrac{1}{2}\pi$.

Feb/March 2021

For each expansion, determine the term that does not involve $x$.

Feb/March 2022

The first term of the geometric progression and the first term of the arithmetic progression are each $a$. The third term of the geometric progression matches the second term of the arithmetic progression. The fifth term of the geometric progression matches the sixth term of the arithmetic progression.

Feb/March 2022

The outside distance around the trunk of a large tree is measured as $5.00\,\text{m}$. One year later, a second measurement gives $5.02\,\text{m}$.

Feb/March 2023

For the expansion of $\left(\frac{x}{a} + \frac{a}{x^2}\right)^7$, you are told that the coefficient of $x^4$ divided by the coefficient of $x$ is $3$.

Feb/March 2023

In the expansion of $(2 + ax)^4(5 - ax)$, the coefficient of $x^3$ is $432$.

Feb/March 2024

In an arithmetic progression whose first term is $6$ and whose tenth term is $19.5$, determine the sum of the first $100$ terms of this arithmetic progression.

Feb/March 2024

Obtain the full expansion of $(2x - \frac{3}{x})^4$.

Feb/March 2025

An arithmetic progression starts with term $5$ and has common difference $6$.

Feb/March 2025

A geometric progression has second term $-120$ and sum to infinity $160$.

Feb/March 2025

Find the first $3$ terms of the expansion of $(2x - \frac{3}{x})^5$ in descending powers of $x$.

May/June 2010

In an arithmetic progression, the 9th term has value $22$, while the sum of the first $4$ terms is $49$. The $n$th term of the progression is $46$.

May/June 2010

Find the first $3$ terms of the expansion of $(1+ax)^5$ in ascending powers of $x$.

May/June 2010

Find the total of all the multiples of $5$ from $100$ to $300$ inclusive.

May/June 2010

The initial term in a geometric progression is 12, and the next term is $-6$.

May/June 2010

State the first three terms, arranged from highest to lowest power of $x$, in the expansion of $(x - \frac{2}{x})^6$.

May/June 2010

Determine the coefficient of $x$ in the expansion of $(x + \frac{2}{x^2})^7$.

May/June 2011

A circle is split into $6$ sectors so that the sector angles form an arithmetic progression. The angle of the largest sector is $4$ times the angle of the smallest sector. If the circle has radius $5\,\text{cm}$, determine the perimeter of the smallest sector.

May/June 2011

Determine the $x^2$ and $x^3$ terms in the expansion of $(1 - \tfrac{3}{2}x)^6$.

May/June 2011

In $(a + x)^5 + (1 - 2x)^6$, the coefficient of $x^3$ equals $90$, with $a$ positive. Find the value of $a$.

May/June 2011

A geometric progression has third term $20$ and sum to infinity equal to three times the first term. Find the first term.

May/June 2011

Find the coefficient of $x^6$ in the expansion of $(2x^3 - \frac{1}{x^2})^7$.

May/June 2012

The first two terms of an arithmetic progression are $1$ and $\cos^2 x$ respectively. Show that the total of the first ten terms can be written in the form $a - b\sin^2 x$, where the constants $a$ and $b$ are to be determined.

May/June 2012

The coefficient of $x^3$ in $(a + x)^5 + (2 - x)^6$ is $90$. Determine the value of the positive constant $a$.

May/June 2012

For an arithmetic progression with the sum of the first $n$ terms, $S_n$, given by $S_n = n^2 + 8n$, determine the first term and the common difference.

May/June 2012

When $(1 - 2x)^2(1 + ax)^6$ is expanded in ascending powers of $x$, the first three terms are $1 - x + bx^2$.

May/June 2012

An arithmetic progression has first term $12$, and the total of its first $9$ terms is $135$.

May/June 2012

In $(1 - px)^6$, $p$ is a non-zero constant. Find the first three terms of the expansion of $(1 - px)^6$ in ascending powers of $x$.

May/June 2013

The third term in a geometric progression is $-108$, while the sixth term is $32$.

May/June 2013

In an arithmetic progression, the first and last terms are $12$ and $48$ respectively, and the sum of the first four terms is $57$. Find the number of terms in the progression.

May/June 2013

Find the coefficient of $x^2$ in the expanded form of $(2x - \frac{1}{2x})^6$.

May/June 2013

Find the first three terms of the expansion of $(2 + ax)^5$ in ascending powers of $x$.

May/June 2013

In an arithmetic progression, the sum $S_n$ of the first $n$ terms is $S_n = 2n^2 + 8n$. Determine the first term and the common difference of the progression.

May/June 2013

Determine the term independent of $x$ in the expansion of $(4x^3 + \frac{1}{2x})^8$.

May/June 2014

For an arithmetic progression with first term $a$ and common difference $d$, it is stated that the sum of the first $200$ terms is $4$ times the sum of the first $100$ terms.

May/June 2014

Find the coefficient of $x^2$ when $(1 + x^2)\left(\frac{x}{2} - \frac{4}{x}\right)^6$ is expanded.

May/June 2014

The 1st, 2nd and 3rd terms of a geometric progression are, in order, the 1st, 9th and 21st terms of an arithmetic progression. The 1st term of each progression is $8$, and the common ratio of the geometric progression is $r$, where $r \neq 1$.

May/June 2014

Determine the coefficient of $x$ in the expansion of $(x^2 - \frac{2}{x})^5$.

May/June 2014

The progression starts with $36$ as the first term and $32$ as the second term.

May/June 2014

Find the first three terms, in ascending powers of $x$, of the expansion of $(1 - x)^6$.

May/June 2015

A geometric progression has third term $\frac{1}{3}$ and fourth term $\frac{2}{9}$ respectively. Find the sum to infinity of the progression.

May/June 2015

Find the coefficients of $x^2$ and $x^3$ within the expansion of $(2 - x)^6$.

May/June 2015

In an arithmetic progression, the first term, second term and final term are $56$, $53$ and $-22$ respectively. Find the sum of all the terms in the progression.

May/June 2015

Give the first 4 terms of the expansion of $(a - x)^5$, arranged in ascending powers of $x$.

May/June 2015

The initial term of an arithmetic progression is $-2222$ and the common difference is $17$. Find the value of the first positive term.

May/June 2015

Determine the term independent of $x$ in the expansion of $(x - \frac{3}{2x})^6$.

May/June 2016

In a geometric progression where every term is positive, the first term is $50$ and the third term is $32$. Find the sum to infinity of this progression.

May/June 2016

Determine the term independent of $x$ in the expansion of $(x - \frac{2}{x})^6$.

May/June 2016

A full water tank contains $2000$ litres. A tiny hole in the bottom is enlarging slowly, so the quantity of water escaping each day is increasing.

May/June 2016

Determine the coefficient of $x$ in the expansion of $(\frac{1}{x} + 3x^2)^5$.

May/June 2016

The $1$st, $3$rd and $13$th terms in an arithmetic progression are also, in that same order, the $1$st, $2$nd and $3$rd terms of a geometric progression. In both progressions, the first term is $3$.

May/June 2016

The coefficients of $x^2$ and $x^3$ in the expansion of $(3 - 2x)^6$ are $a$ and $b$ respectively.

May/June 2017

An arithmetic progression begins with $32$, has a 5th term of $22$ and ends with $-28$. Determine the sum of every term in the progression.

May/June 2017

Determine the coefficient of $x$ in the expansion of $(2x - \frac{1}{x})^5$.

May/June 2017

The first two terms of an arithmetic progression are $16$ and $24$. Determine the smallest number of terms from the progression that need to be included so that their sum is greater than $20\,000$.

May/June 2017

Within the expansion of $(2 + ax)^7$, the coefficients of $x$ and $x^2$ are equal.

May/June 2017

For a geometric progression with common ratio $r$, the first term is $(r^2 - 3r + 2)$ and the infinite sum is $S$.

May/June 2017

Find the first three terms, written in ascending powers of $x$, in the expansion of $(1 - 2x)^5$.

May/June 2018

Find the possible values of the first term of a geometric progression with second term $12$ and sum to infinity $54$.

May/June 2018

In the expansion of $(2 + \frac{x}{2})^6 + (a + x)^5$, the coefficient of $x^2$ comes to $330$.

May/June 2018

A firm that made salt from sea water switched to a different process. Each week, the quantity of salt produced rose by $2\%$ of the amount produced in the previous week. It is stated that during the first week after the change the firm produced $8000\text{ kg}$ of salt.

May/June 2018

Determine the coefficient of $\frac{1}{x}$ in the expansion of $\left(x - \frac{2}{x}\right)^5$.

May/June 2018

The common ratio in the geometric progression is $0.99$. Write the sum of the first $100$ terms as a percentage of the sum to infinity, and give the result correct to $2$ significant figures.

May/June 2018

In the expansion of $(2x + \frac{k}{x})^6$, where $k$ is a constant, the term independent of $x$ is $540$.

May/June 2019

In a geometric progression, the 3rd and 4th terms are $48$ and $32$ respectively. Find the sum to infinity of the progression.

May/June 2019

Find the coefficient of $x$ when $\left(\frac{2}{x} - 3x\right)^5$ is expanded.

May/June 2019

In an arithmetic progression, the total of the first ten terms is the same as the total of the next five terms. The first term is $a$.

May/June 2019

The first three terms in the binomial expansion of $(2x - \frac{1}{2x})^5$ are $32x^5 - 40x^3 + 20x$. Find the other three terms of the expansion.

May/June 2019

Two heavyweight boxers decide that they would achieve better results if they entered a lighter weight class. For each boxer, this means a total loss of 13 kg. By the end of week 1, each has lost 1 kg, and in every later week their weight loss is a little smaller than in the week before. Boxer $A$ loses 0.98 kg in week 2. It is stated that his weekly weight loss is in arithmetic progression.

May/June 2019

The first nine terms of an arithmetic progression add to 117. The next four terms add to 91.

May/June 2020

When $(kx + \frac{1}{x})^5 + (1 - \frac{2}{x})^8$ is expanded, the coefficient attached to $\frac{1}{x}$ is 74.

May/June 2020

Each year, the selling price of a diamond necklace rises by 5% compared with the price in the previous year. In the year 2000, the necklace was priced at $36000$.

May/June 2020

Find the coefficient of $x^2$ when $(x - \frac{2}{x})^6$ is expanded.

May/June 2020

The formula for the $n$th term of an arithmetic progression is $\frac{1}{2}(3n - 15)$.

May/June 2020

Expand $(1 + a)^5$ in ascending powers of $a$ as far as and including the term in $a^3$.

May/June 2020

The initial term of the progression is $\sin^2 \theta$, with $0 < \theta < \tfrac{1}{2}\pi$. Its second term is $\sin^2 \theta \cos^2 \theta$.

May/June 2020

The total of the first 20 terms in an arithmetic progression is $405$, while the total of the first 40 terms is $1410$.

May/June 2021

Find the first three terms of the expansion of $(3 - 2x)^5$ in ascending powers of $x$.

May/June 2021

The fifth, sixth, and seventh terms in a geometric progression are $8k$, $-12$, and $2k$ respectively.

May/June 2021

In the expansion of $(4x + \frac{10}{x})^3$, the coefficient of $x$ is $p$. In the expansion of $(2x + \frac{k}{x^2})^5$, the coefficient of $\frac{1}{x}$ is $q$. If $p = 6q$, determine the possible values of $k$.

May/June 2021

In an arithmetic progression, the first, second and third terms are $a$, $\frac{3}{2}a$ and $b$ respectively, where $a$ and $b$ are positive constants. In a geometric progression, the first, second and third terms are $a$, $18$ and $b + 3$ respectively.

May/June 2021

Write down the first four terms, arranged in ascending powers of $x$, of the expansion of $(a - x)^6$.

May/June 2021

For a geometric progression, the second term is $24\%$ of the sum to infinity. Determine the possible values of the common ratio.

May/June 2021

In this arithmetic progression, the 13th term is $12$ and the total of the first $30$ terms is $-15$.

May/June 2022

In $(3 + x)^5$, the coefficient of $x^4$ is the same as the coefficient of $x^2$ in $(2x + \frac{a}{x})^6$.

May/June 2022

In a geometric progression, the second term is $10$ and the third term is $8$.

May/June 2022

In an arithmetic progression, the first three terms are $k$, $6k$ and $k + 6$ in that order.

May/June 2022

An arithmetic progression starts with term $4$ and has common difference $d$. The total of its first $n$ terms is $5863$.

May/June 2022

Find the first three terms in the expansion, in ascending powers of $x$, of $(2 + 3x)^4$.

May/June 2023

The initial three members of an arithmetic progression are $\frac{p^2}{6}$, $2p - 6$ and $p$.

May/June 2023

In the expansion of $(x + a)^6$, the coefficient of $x^4$ is $p$, and in the expansion of $(ax + 3)^4$, the coefficient of $x^2$ is $q$. You are told that $p + q = 276$.

May/June 2023

The second term in a geometric progression is $16$, and the sum to infinity is $100$.

May/June 2023