Arithmetic Sequences & Series (AQA A Level Maths: Pure): Exam Questions

An arithmetic sequence is defined as follows:

• The third term is $32$

• The eleventh term is $0$

• The sum of the first $n$ terms is $-44$

Find the value of $n$.

The first three terms in an arithmetic sequence are 

$(q-2),q^2,(4q+5),\dots$

Find the possible values of $q$.

The 4^th and 8^th terms of an arithmetic sequence are 20 and 64 respectively.

Find the first term, $a$, and the common difference, $d$.

The 12^th and 16^th terms of a different arithmetic sequence differ by 20.

Find the possible values of the common difference.

Prove that the sum of the first $n$ odd numbers is a square number, for any positive integer value of $n$.

An arithmetic sequence is defined as follows:

• The first term is 3

• The 10^th term is 30

• The sum of the first $n$ terms is 630

 Find the common difference.

Show that $n$ satisfies the equation

$n^2+n-420=0$

Hence find the value of $n$.

An arithmetic sequence is defined as follows:

• The third term is 2

• The twelfth term is 65

• The sum of the first $n$ terms is 390

Show that 

$7n^2-31n-780=0$

Hence find the value of $n$.

The sum of the first ten terms in an arithmetic series is 40. 

The sum of the first twenty terms in the same series is 280. 

Find the first term, $a$, and the common difference, $d$, of the series.

The first three terms in an arithmetic sequence are 

$(p-9),-7,(9-3p),\dots$

Find the value of $p$.

The first three terms in an arithmetic sequence are 

$-1,(q+1),q^2,\dots$

Find the possible values of $q$.

An arithmetic sequence is given by

$k,2k,3k,4k,...$

where $k$ is a constant.

Find a formula for the $n$^th term of the sequence, giving your answer in terms of $k$ and $n$.

Show that the sum of the first $n$ terms is given by

$\frac{kn}{2}\left(n+1\right)$

The sum of the first 12 terms is 39.

Find the value of $k$.

Find the sum of all the odd numbers between 0 and 150:

        $1+3+5+\dots +149$

An arithmetic sequence is given by

$(3q-7),(5q-4),(7q-1),\dots$

where $q$ is a constant.

Find an expression for the 100^th term of the sequence, giving your answer the form

$bq+c$

where $b$ and $c$ are integers to be found.

The first two terms in an arithmetic sequence are $(p+15)$ and 3. 

The fourth term is $(3p-16)$.

Find the value of $p$.

An arithmetic sequence has

• a first term of $r^2$, where $r>0$

• a common difference of $3r$

• a fifth term of 85

Find the ninth term in the sequence.

An arithmetic sequence is defined in terms of a constant, $k$, as follows:

• The fifth term is $k$

• The sum of the first eight terms is $2k$

Show that the first term of the sequence is $-5k$

Find an expression for the common difference of the sequence in terms of $k$ only.

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

Find the sum of the first 30 terms of the sequence.

The sum of the first twelve terms in an arithmetic series is 654.

The sum of the first twenty terms in the same arithmetic series is 530.

Find the 21^st term in the sequence.

Prove that the sum of the first $n$ terms of an arithmetic series is given by

$S_n=\frac{n}{2}\left[2a+\left(n-1\right)d\right]$

An arithmetic series $u_n$ has first term $a=10$.

Given that $\sum _{n=21}^{50}{u}_n=3150$ find the value of the common difference, $d$.

An arithmetic sequence is defined in terms of a constant, $k$, as follows:

• The seventh term is $3k$

• The sum of the first nine terms is $\left(4k-3\right)$

Show that the common difference is $\frac{23k+3}{18}$ and find an expression for the first term of the sequence, giving your answer as a single simplified algebraic fraction in terms of $k$.

Given that the 19^th term of the sequence is 57, find the sum of the first 25 terms.

The first four terms in an arithmetic sequence are 

$(2q-p),1,(p-q),(3p-4q),\dots$

Find the values of $p$ and $q$.

An arithmetic sequence is given by

$\left(k+7\right),\left(2k+18\right),\left(3k+29\right),...$

where $k$ is a constant.

If the $n$^th term of the sequence is 36, show that

$n=\frac{40}{k+11}$

Hence show that the sum of the first $n$ terms of the sequence is

 $\frac{20k+860}{k+11}$

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

The first three terms in an arithmetic sequence are 

$2,3q^2,q,\dots$

Given that these first three terms are all positive, find the value of the 40^th term in the sequence.

The sum of the first 24 terms in an arithmetic series is nine times the sum of the first two terms in the series.

Find the sum of the first 90 terms in the series.

An arithmetic sequence is given by

            $(-21+\ln 3),(-20+\ln 27),(-19+\ln 243),\dots$

The sum of the first $n$ terms of the sequence is

$pn+q{n}^2$

where $p$ is an exact rational constant and $q$ is an exact irrational constant.

Find $p$ and $q$, giving your answers in simplified form

Hence find the smallest value of $n$ for which the sum of the first $n$ terms is greater than zero.

An arithmetic sequence has

• a first term of $r^2$, where $r>0$

• a common difference of $r$

• a ninth term of 7

$S$ is the set of all the numbers that are in the sequence. 

$T$ is the set of all rational numbers.

Find all the elements in the set $S\cap T$.

The $k$^th term of an arithmetic series is 0.

The sum of the first $n$ terms is also 0.

The common difference of the series is not equal to 0.

Find an expression for $n$ in terms of $k$ only, showing your working clearly.

For an arithmetic series

• the sum of the first 20 terms is 290

• the sum of the first 24 terms is -180

• the sum of the first  $n$ terms is $S_n$

Find the greatest value that $S_n$ can take, for any positive integer $n$.

The $n$th term in an arithmetic sequence is

$\left(n+\frac{1}{2}\right)\ln 4$

Show that the sum of the first $n$ terms of the sequence is given by

$\ln \left(2^{\mathrm{f}\left(n\right)}\right)$

where $\mathrm{f}\left(n\right)$ is a polynomial to be found, in terms of $n$.