Arithmetic Progressions (Cambridge International A-Level Mathematics): Exam Questions

Write down the next three terms in these arithmetic sequences

(i) $30 , 18 , 6 , \dots$

(ii) $\frac{1}{4} , \frac{5}{12} , \frac{7}{12} , \frac{3}{4} , \dots$

Find the sum of the first four terms in the sequence defined by $u_{n} = 2 n + 3$. Justify why this sequence is an arithmetic sequence.

Write down a formula for the $n$th term of each of the following arithmetic sequences

(i) $16 , 20 , 24 , \dots$

(ii) First term $a = 3$, common difference $d = - 3$

(iii) $a = 2 , d = 6$

Find the 10th and 20th terms in each of the following arithmetic sequences

(i) $u_{n} = 4 + 5 n$

(ii) $u_{n} = \frac{1}{2} - \frac{1}{4} n$

(iii) $u_{n} = 50 - 5 n$

An arithmetic progression has fourth term $20$ and eighth term $64$.

Find the first term and the common difference of the progression.

A different arithmetic progression is such that its twelfth term and its sixteenth term differ by $20$.

Find the possible values of the common difference of the progression.

An arithmetic progression has first term $3$ and common difference $4$.

Find the sum of the first $20$ terms of the progression.

An arithmetic progression has first term $3$ and tenth term $30$. The sum of the first $n$ terms of the progression is $630$.

Find the common difference of the progression.

Show that $n^{2} + n - 420 = 0$.

Hence find the value of $n$.

The first three terms of an arithmetic progression are $k$, $2 k$ and $3 k$, where $k$ is a non-zero constant.

Write down a formula, in terms of $n$ and $k$, for the $n$th term of the progression.

Show that the sum of the first $n$ terms of the progression is $\frac{kn}{2} ( n + 1 )$.

Given that the sum of the first $12$ terms of the progression is $39$, find the value of $k$.

The first three terms of an arithmetic progression are $p - 9$, $- 7$ and $9 - 3 p$ respectively.

Find the value of $p$.

The first three terms of an arithmetic progression are $- 1$, $q + 1$ and $q^{2}$ respectively.

Find the possible values of $q$.

An arithmetic progression has first term $r^{2}$ and common difference $3 r$, where $r > 0$. The fifth term of the progression is $85$.

Find

(i) the value of $r$,

(ii) the ninth term of the progression.

An arithmetic progression has third term $2$ and twelfth term $65$. The sum of the first $n$ terms of the progression is $390$.

Show that $7 n^{2} - 31 n - 780 = 0$.

Hence find the value of $n$.

The sum of the first ten terms of an arithmetic progression is $40$. The sum of the first twenty terms of the same progression is $280$.

Find the first term and the common difference of the progression.

The first three terms of an arithmetic progression are $k + 7$, $2 k + 18$ and $3 k + 29$. The $n$th term is $36$.

Show that $n = \frac{40}{k+11}$.

Hence show that the sum of the first $n$ terms is $\frac{20k+860}{k+11}$.

Given that the sum of the first $n$ terms is $180$, find the value of $k$.

The fifth term of an arithmetic progression is $k$, where $k$ is a constant, and the sum of the first eight terms of the progression is $2 k$.

Show that the first term, $a$, of the progression is $- 5 k$.

Find an expression for the common difference, $d$, of the progression in terms of $k$.

Given that the ninth term of the progression is $14$, find the value of $k$.

Find the sum of the first thirty terms of the progression.

The $k$th term of an arithmetic progression is given by $u_{k} = 26 - 3 k$.

Calculate the sum of the first ten terms of the progression.

Calculate the sum of the eleventh to fifteenth terms of the progression, inclusive.

Calculate the sum of all the odd numbers between $0$ and $150$:

$1 + 3 + 5 + \dots + 149$

Below is an arithmetic progression

$k + 2 k + 3 k + \dots + 360$

where $k$ is an integer and a positive factor of $360$.

(i) In terms of $k$, find an expression for the number of terms in the progression.

(ii) Show that the sum of the terms in the progression is $180 + \frac{64800}{k}$.

The first three terms of an arithmetic progression are $3 q - 7$, $5 q - 4$ and $7 q - 1$.

In terms of $q$, find the $100$th term of the progression, giving your answer in simplest form.

The first two terms of an arithmetic progression are $p + 15$ and $3$. The fourth term of the progression is $3 p - 16$.

Find the value of $p$.

The first three terms of an arithmetic progression are $q - 2$, $q^{2}$ and $4 q + 5$ respectively.

Find the possible values of $q$.

An arithmetic progression has first term $r^{2}$ and common difference $2 r$, where $r > 0$. The fourth term of the progression is $4$.

Find the value of $r$, giving your answer as an exact value.

The sum of the first twelve terms of an arithmetic progression is $654$. The sum of the first twenty terms of the same progression is $530$.

Find the twenty-first term of the progression.

The $k$th term of an arithmetic progression is given by $u_{k} = 16 + 7 k$.

Calculate the sum of the fifteenth to twenty-fifth terms of the progression, inclusive.

Calculate the sum of the first $25$ terms of the progression whose $k$th term is given by $v_{k} = u_{k} - 3$, where $u_{k}$ is defined as above.

The first three terms of an arithmetic progression are $2$, $3 q^{2}$ and $q$ respectively.

Given that the first three terms are all positive, find the fortieth term of the progression.

An arithmetic progression has third term $32$ and eleventh term $0$. The sum of the first $n$ terms of the progression is $- 44$.

Find the value of $n$.

Show that the sum of the first $n$ odd numbers $1 + 3 + 5 + \dots$ is $n^{2}$, for any integer $n \geq 1$.

The first three terms of an arithmetic progression are $7 k$, $14 k$ and $21 k$, where $k$ is an integer. The $n$th term is $1008$.

(i) In terms of $k$, find an expression for the number of terms in the progression.

(ii) In addition to being an integer, state the two other conditions that $k$ must satisfy for this to be a valid arithmetic progression.

(iii) Show that the sum of the first $n$ terms of the progression is $504 + \frac{72576}{k}$.

The seventh term of an arithmetic progression is $3 k$, where $k$ is a constant, and the sum of the first nine terms of the progression is $4 k - 3$.

In terms of $k$, find expressions for the first term and the common difference of the progression.

Given that the nineteenth term of the progression is $57$, find the sum of the first $25$ terms of the progression.

The first three terms of an arithmetic progression are $k + 13$, $2 k + 9$ and $3 k + 5$. The $n$th term of the progression is denoted by $u_{n}$.

Find an expression for $n$ in terms of $u_{n}$ and $k$.

For a particular value of $n$, $u_{n} = - 16$ and the sum of the first $n$ terms of the progression is $- 11$.

Find the value of $k$.

The first four terms of an arithmetic progression are $2 q - p$, $1$, $p - q$ and $3 p - 4 q$ respectively.

Find the values of $p$ and $q$.

The $k$th term of an arithmetic progression is given by $u_{k} = 89 - 5 k$.

Given that the sum of the first $n$ terms is equal to $- 35$, find the value of $n$.

The sum of the first $24$ terms of an arithmetic progression is nine times the sum of the first two terms of the progression.

Find the sum of the first $90$ terms of the progression.

An arithmetic progression has first term $a$ and common difference $d$.

Given that

• the sum of the seventh to twelfth terms, inclusive, is $- 69$,
• the sum of the seventh to sixteenth terms, inclusive, is $- 175$, and
• the sum of the first six terms is $- 13 d$,

find the values of $a$ and $d$.

The $k$th term of an arithmetic progression is $0$. The sum of the first $n$ terms is also $0$.

Find the value of $n$ in terms of $k$, giving clear reasons for your answer.

The sum of the first $20$ terms of an arithmetic progression is $290$. The sum of the first $24$ terms of the same progression is $- 180$. The sum of the first $n$ terms is denoted by $S_{n}$.

Find the greatest value attained by $S_{n}$ for any $n \geq 1$.