Browse Sequences & series

136 questions at your level

DifficultyAll Level 1 Level 2 Level 3 Level 4 Level 5

The binomial expansion

11 questions

LessonNot started

Find the term which is independent of

x

in the expansion of

\left(2x - \dfrac{3}{x^{2}}\right)^{6}

●●●●●Level 24 marksStart

Arithmetic sequences and series

38 questions

LessonNot started

The third term of an arithmetic sequence is

13

and the seventh term is

29

.
(a) Find the first term and the common difference. (2 marks)
(b) Find the sum of the first

20

terms. (2 marks)

●●●●●Level 24 marksStart

The third term of an arithmetic sequence is

13

and the seventh term is

29

.

Find the sum of the first

20

terms of the sequence. (4 marks)

●●●●●Level 24 marksStart

(a) Show that

\displaystyle\sum_{r=1}^{n} (6r + 2) = 3n^2 + 5n

. (2 marks)
(b) Hence determine the value of

n

\displaystyle\sum_{r=1}^{n} (6r + 2) = 100

●●●●●Level 25 marksStart

3, 7, 11, 15, \dots

is arithmetic.
(a) Find an expression for the

n

th term.
(b) Hence find the

50

●●●●●Level 23 marksStart

An arithmetic sequence has first term

8

and common difference

5

.
(a) Find an expression for the

n

th term. (2 marks)
(b) Find the

20

th term. (1 mark)

●●●●●Level 23 marksStart

Find the sum of the first

30

terms of the arithmetic series

2 + 9 + 16 + 23 + \cdots

●●●●●Level 23 marksStart

n

th term of an arithmetic sequence is

u_{n} = 4n - 1

.

Find the value of

n

u_{n} = 199

●●●●●Level 23 marksStart

Geometric sequences and series

38 questions

LessonNot started

2, 6, 18, 54, \dots

is geometric.
(a) Find an expression for the

n

th term.
(b) Hence find the

6

●●●●●Level 23 marksStart

Sigma notation and recurrence relations

29 questions

LessonNot started

u_1, u_2, u_3, \ldots

u_1 = 3

u_{n+1} = \frac{u_n + 1}{u_n - 1}

(a) (i) Show that this sequence is periodic.

(ii) State the order of this periodic sequence.

(2 marks)
(b) Find the value of

\displaystyle\sum_{n=1}^{75} u_n

●●●●●Level 24 marksStart

u_1, u_2, u_3, \ldots

u_1 = k

u_{n+1} = 8 - u_n

k

is a constant.
(a) Find, in terms of

k

u_2

u_3

(2 marks)

Given that

\sum_{n=1}^{41} u_n = 173

(b) find the value of

k

●●●●●Level 24 marksStart

(a) Show that

\displaystyle\sum_{r=1}^{n} (6r + 2) = 3n^2 + 5n

. (2 marks)
(b) Hence determine the value of

n

\displaystyle\sum_{r=1}^{n} (6r + 2) = 100

●●●●●Level 25 marksStart

A sequence is defined by

u_{1} = 5

u_{n+1} = u_{n} + 4

.
(a) Write down

u_{2}

u_{3}

u_{4}

. (2 marks)
(b) Find the value of

n

u_{n} = 101

●●●●●Level 24 marksStart

Binomial expansion for any index

35 questions

LessonNot started

(a) Find the first three terms, in ascending powers of

x

, of the binomial expansion of

\sqrt{9 - 8x}

writing each term in simplest form.

(3 marks)

A student uses this expansion with

x = \dfrac{1}{4}

to find an approximation for

\sqrt{7}

Using the answer to part (a) and without doing any calculations,
(b) state whether this approximation will be an overestimate or an underestimate of

\sqrt{7}

, giving a brief reason for your answer.

(1 mark)

●●●●●Level 24 marksStart

(a) Find the first three terms, in ascending powers of

x

, of the binomial expansion of

(8 - 3x)^{\frac{1}{3}}

writing each term in simplest form.

(3 marks)

A student attempts to use this expansion with

x = -3

to find an approximation for

\sqrt[3]{17}

(b) Give a reason why

x = -3

should not be used in this expansion to find an approximation for

\sqrt[3]{17}

●●●●●Level 24 marksStart

(a) Find the first three terms, in ascending powers of

x

, of the binomial expansion of

(8+3x)^{\frac{2}{3}}

,

and state the range of values of

x

for which the expansion is valid. (4 marks)
(b) Use your expansion to estimate

8.3^{\frac{2}{3}}

, giving your answer to 4 decimal places. (1 mark)

●●●●●Level 25 marksStart