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

A running club has 169 members who run mainly on roads and 91 members who run mainly on trails. A random sample of 18 members is selected to attend a meeting about the club's new training programme. The number of members in the sample who run mainly on trails is denoted by

X

.
(a) State a suitable binomial model for

X

.

(1 mark)

Use your model to answer the following.
(b) (i) Write down an expression for

\mathrm{P}(X = x)

.

(ii) State, in set notation, the values of

x

for which your expression is valid.

(iii) Explain the connection between the expression in part (b)(i) and the binomial expansion of

(0.65 + 0.35)^n

, for a specific value of

n

which should be stated.

(4 marks)
(c) Find

\mathrm{P}(4 \leqslant X \leqslant 8)

.

(2 marks)
(d) State one disadvantage of using a random sample in this context.

(1 mark)

●●●●●Level 48 marksStart

In this question you must show detailed reasoning.

The functions f and g are defined for all real values of

x

f(x) = x^3

g(x) = x^2 - 4

.
(a) Write down expressions for

(i)

fg(x)

gf(x)

.

(2 marks)
(b) Hence find the values of

x

fg(x) - gf(x) = -120

●●●●●Level 48 marksStart

(a) Find the first four terms in the expansion of

(2 - x)^6

in ascending powers of

x

.

(4 marks)
(b) Hence determine the coefficient of

y^3

in the expansion of

(2 - y - 3y^2)^6

●●●●●Level 48 marksStart

\text{f}(x) = (3 + kx)^8 \qquad \text{where } k \text{ is a non-zero constant}

(a) Find the first 4 terms, in ascending powers of

x

, of the binomial expansion of

\text{f}(x)

. Give each term in simplest form.

(4 marks)

Given that, in the binomial expansion of

\text{f}(x)

, the constant term, the coefficient of

x

and the coefficient of

x^2

are the first 3 terms of an arithmetic progression,
(b) find, using algebra, the possible values of

k

.

(Solutions relying entirely on calculator technology are not acceptable.)

(3 marks)

●●●●●Level 47 marksStart

The binomial expansion, in ascending powers of

x

(4 + px)^5

p

is a constant, can be written in the form

A + Bx + Cx^2 + Dx^3 + \ldots

A

B

C

D

are constants.
(a) Find the value of

A

(1 mark)

Given that

$B = 32D$
$p < 0$

(b) find

(i) the value of

p

(ii) the value of

D

●●●●●Level 47 marksStart

(a) Find, in terms of

a

, the first 3 terms, in ascending powers of

x

, of the binomial expansion of

(3 + ax)^5

a

is a positive constant. Give each term in simplest form.

(3 marks)

\text{f}(x) = \left(2x - \frac{1}{x}\right)^2(3 + ax)^5

Given that the constant term in the expansion of

\text{f}(x)

108

(b) find the value of

a

●●●●●Level 47 marksStart

Arithmetic sequences and series

38 questions

LessonNot started

A fruit grower plants a new orchard. In its first year the orchard yields

1850

kg of apples. The grower assumes that the yield will go up by the same amount every year. In the second year the orchard yields

1940

kg of apples.
(a) Find the yield of the orchard in the twelfth year.

(2 marks)
(b) Find the number of complete years needed for the total yield of the orchard to first exceed

50\,000

kg.

(4 marks)
(c) Comment on how realistic this model may be in the long term.

(1 mark)

●●●●●Level 47 marksStart

Nadia pays money into a savings account once a month. She pays in

\pounds a

in the first month, and each month after that she pays in

\pounds d

more than in the month before. In the fifth month she pays in

\pounds 78

. The total amount that she pays in over the first

12

\pounds 1062

.
(a) Determine the values of

a

d

.

(4 marks)
(b) Find the number of complete months needed for the total amount that Nadia has paid in to first exceed

\pounds 5000

.

(3 marks)
(c) Give a reason why this model may not be appropriate in the long term.

(1 mark)

●●●●●Level 48 marksStart

The first three terms of an arithmetic progression are the positive integers

x

y

z

, in that order. The common difference of this progression is

-3

x

\dfrac{20}{y}

z

are the first, second and third terms, respectively, of a geometric progression.
(a) Show that

y

satisfies the equation

y^4 - 9y^2 - 400 = 0

.

(4 marks)
(b) Hence determine the sum to infinity of the geometric progression.

(4 marks)

●●●●●Level 48 marksStart

The first, second and fourth terms of an arithmetic progression are

u_1

u_2

u_4

respectively, where

u_1 = 3\sin\theta, \qquad u_2 = 5\cos\theta, \qquad u_4 = -21\sin\theta

\tfrac12\pi < \theta < \pi

.
(a) Determine the exact value of

\theta

.

(3 marks)
(b) Hence determine the value of

\displaystyle\sum_{r=1}^{20} u_r

●●●●●Level 46 marksStart

Two arithmetic progressions,

P

Q

60

terms, denoted by

p_i

q_i

respectively, where

i = 1, 2, 3, \ldots, 60

.

The common difference of

P

d

d

is a positive integer. The progressions are related as follows.

$p_1 = q_{60}$

$q_1 = p_{60}$

(a) You are given that there is at least one value of

i

q_i = p_i + 24

.

Show that, in this case,

i = \dfrac{61}{2} - \dfrac{12}{d}

.

(6 marks)
(b) Hence show that it is impossible for the equation

q_i = p_i + 24

d

takes certain values, which should be stated.

(2 marks)

●●●●●Level 48 marksStart

A small business makes handmade candles.

In year 1, the business made $1500$ candles.
In year 10, the business is predicted to make $5400$ candles.

Model A assumes that the number of candles made will increase by the same amount each year.
(a) Using model A, find the number of candles made in year 5. Give your answer to the nearest 10 candles.

(3 marks)

Model B assumes that the number of candles made will increase by the same percentage each year.
(b) Using model B, find the number of candles made in year 3. Give your answer to the nearest 10 candles.

(3 marks)
(c) Calculate, according to the two models, the difference between the total numbers of candles predicted to be made from year 1 to year 10 inclusive. Give your answer to the nearest 10 candles.

(3 marks)

●●●●●Level 49 marksStart

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.
(a) A geometric series begins

45 + 30 + 20 + \ldots

(i) Find the sum to infinity of this series.

(ii) Given that the sum to infinity of this series exceeds the sum of the first

n

terms by less than

0.05

, use algebra to find the smallest possible value of

n

.

(5 marks)
(b) An arithmetic series begins

196 + 190.5 + 185 + \ldots

Given that the sum of the first

n

terms of this series is

S_n

, find the greatest possible value of

S_n

●●●●●Level 49 marksStart

\text{f}(x) = (3 + kx)^8 \qquad \text{where } k \text{ is a non-zero constant}

(a) Find the first 4 terms, in ascending powers of

x

, of the binomial expansion of

\text{f}(x)

. Give each term in simplest form.

(4 marks)

Given that, in the binomial expansion of

\text{f}(x)

, the constant term, the coefficient of

x

and the coefficient of

x^2

are the first 3 terms of an arithmetic progression,
(b) find, using algebra, the possible values of

k

.

(Solutions relying entirely on calculator technology are not acceptable.)

(3 marks)

●●●●●Level 47 marksStart

An arithmetic sequence has first term

a

and non-zero common difference

d

2

5

11

th terms of the arithmetic sequence form the first three terms of a geometric sequence.
(a) Show that the common ratio of the geometric sequence is

2

. (5 marks)
(b) Given that

a = 4

, find the sum of the first

20

terms of the arithmetic sequence. (2 marks)

●●●●●Level 47 marksStart

The first three terms of a geometric progression are

4k

k+3

k

k

is a non-zero constant.
(a) Show that

k^{2} - 2k - 3 = 0

and hence find the two possible values of

k

. (4 marks)
(b) It is given that the common ratio of the progression is positive. Find the sum to infinity. (2 marks)
(c) An arithmetic progression has first term equal to the first term of the geometric progression in part (b), and common difference

5

. Find the least number of terms whose sum exceeds

500

●●●●●Level 49 marksStart

The first, second and fifth terms of an arithmetic sequence, with first term

a

and common difference

d

d \neq 0

, are also the first three terms of a geometric sequence.

The sum of the first

20

terms of the arithmetic sequence is

800

.

Find the smallest value of

n

n

th term of the geometric sequence is greater than one million. (6 marks)

●●●●●Level 46 marksStart

A charity receives a donation at the end of each week. The donation at the end of week

1

120

, and each week the donation is £

45

more than the donation the week before.
(a) Find the donation received at the end of week

30

. (2 marks)
(b) The charity's target is a total of £

50\,000

. Find the number of the week at the end of which the running total first exceeds £

50\,000

●●●●●Level 45 marksStart

A geometric sequence

u_1, u_2, u_3, \ldots

u_1 = 2

and common ratio

4

.
(a) Show that

\displaystyle\sum_{r=1}^{n} \log_2 u_r = n^2

. (3 marks)
(b) Hence determine the smallest value of

n

u_1 \times u_2 \times u_3 \times \cdots \times u_n > 10^{30}

●●●●●Level 47 marksStart

The sum of the first

n

terms of a series is

S_{n} = 3n^{2} + 2n

.
(a) Find the first term. (1 mark)
(b) Show that the series is arithmetic, and find an expression for the

n

th term and the common difference. (4 marks)

●●●●●Level 45 marksStart

Priya is training for a long-distance race.

In week

1

she runs a total of

12

km. Each week she runs

2

km further than she ran the week before, so that the weekly distances form an arithmetic sequence.
(a) Show that Priya runs

30

10

.

(1 mark)

Given that after

n

weeks Priya has run a total of exactly

350

km,
(b) show that

n^2 + 11n - 350 = 0

(2 marks)
(c) Hence find the value of

n

●●●●●Level 45 marksStart

Geometric sequences and series

38 questions

LessonNot started

Nadia and Owen are playing a game.

The game consists of four rounds.
In each round, $\mathrm{P}(\text{Nadia wins the round}) = 0.75$ and $\mathrm{P}(\text{Owen wins the round}) = 0.25$ .
The outcome of each round is independent of the outcome of every other round.
If one player wins at least three of the four rounds, then they win the game, otherwise neither player wins.

(a) Determine the probability of the following events.

(i) Nadia wins the game.

(ii) Neither player wins the game.

(5 marks)

Nadia and Owen now decide to play a match which consists of a series of games.

In each game, if a player wins the game then they win the match.
If neither player wins the game then the players play another game.

(b) Determine the probability that Owen wins the match.

(3 marks)

●●●●●Level 48 marksStart

A sculptor builds a tower from wooden blocks, placing each block on top of the one before. The bottom block is

8

cm tall, and each block above it is

9\%

shorter than the block directly beneath it.

The finished tower is

60

cm tall, correct to the nearest centimetre.

Determine the number of blocks in the tower.

(5 marks)

●●●●●Level 45 marksStart

Imogen is learning to build bicycle wheels. Her first wheel takes her

84

minutes to build. Her second wheel takes her

75.6

minutes to build. Imogen believes that the times she takes can be modelled by a geometric progression.
(a) Imogen sets herself a target of building a wheel in less than

30

minutes. Show that this model predicts that Imogen will achieve her target on her eleventh wheel.

(4 marks)
(b) After some weeks, Imogen has spent a total of

603

minutes, to the nearest minute, building wheels. Use this model to find how many wheels she has built.

(3 marks)
(c) Give two reasons why this model may not be appropriate when predicting the time Imogen takes to build a wheel.

(2 marks)

●●●●●Level 49 marksStart

In this question you must show detailed reasoning.

The first three terms of a convergent geometric progression are

3x + 5

x + 5

2x - 5

respectively.

Determine the sum to infinity of this geometric progression.

(8 marks)

●●●●●Level 48 marksStart

In this question you must show detailed reasoning.

A sequence

u_1, u_2, u_3, \ldots

u_n = 18 \times 0.75^n

.

Use an algebraic method to find the smallest value of

N

\sum_{n=1}^{\infty} u_n - \sum_{n=1}^{N} u_n < 10^{-3}.

●●●●●Level 48 marksStart

The first three terms of an arithmetic progression are the positive integers

x

y

z

, in that order. The common difference of this progression is

-3

x

\dfrac{20}{y}

z

are the first, second and third terms, respectively, of a geometric progression.
(a) Show that

y

satisfies the equation

y^4 - 9y^2 - 400 = 0

.

(4 marks)
(b) Hence determine the sum to infinity of the geometric progression.

(4 marks)

●●●●●Level 48 marksStart

In this question you should assume that $-1 < x < 1$ .
(a) (i) Find the first four terms in the expansion of

(1 + x)^{-2}

in ascending powers of

x

.

(ii) Write down the term in

x^n

in this expansion, where

n

is a positive integer.

(3 marks)
(b) Write down the sum to infinity of the series

1 - x + x^2 - x^3 + \ldots

(1 mark)
(c) Hence or otherwise find and simplify an expression for

2 - 5x + 8x^2 - 11x^3 + \ldots

\dfrac{a + bx}{(1 + x)^2}

a

b

are constants to be determined.

(3 marks)

●●●●●Level 47 marksStart

A small business makes handmade candles.

In year 1, the business made $1500$ candles.
In year 10, the business is predicted to make $5400$ candles.

Model A assumes that the number of candles made will increase by the same amount each year.
(a) Using model A, find the number of candles made in year 5. Give your answer to the nearest 10 candles.

(3 marks)

Model B assumes that the number of candles made will increase by the same percentage each year.
(b) Using model B, find the number of candles made in year 3. Give your answer to the nearest 10 candles.

(3 marks)
(c) Calculate, according to the two models, the difference between the total numbers of candles predicted to be made from year 1 to year 10 inclusive. Give your answer to the nearest 10 candles.

(3 marks)

●●●●●Level 49 marksStart

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

A volunteer group is planting saplings as part of a rewilding project.

The numbers of saplings planted each month, in hundreds, for the first three months of the project were

k - 4 \qquad\qquad k \qquad\qquad 3k - 35

k

is a constant.

Given that the numbers of saplings planted each month are modelled as a geometric series,
(a) show that

2k^{2} - 47k + 140 = 0

(2 marks)
(b) predict the number of saplings planted in the 4th month.

(4 marks)

The total number of saplings planted is predicted to exceed

30\,000

for the first time in the

N

th month.
(c) Calculate the value of

N

according to the model.

(3 marks)

●●●●●Level 49 marksStart

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.
(a) A geometric series begins

45 + 30 + 20 + \ldots

(i) Find the sum to infinity of this series.

(ii) Given that the sum to infinity of this series exceeds the sum of the first

n

terms by less than

0.05

, use algebra to find the smallest possible value of

n

.

(5 marks)
(b) An arithmetic series begins

196 + 190.5 + 185 + \ldots

Given that the sum of the first

n

terms of this series is

S_n

, find the greatest possible value of

S_n

●●●●●Level 49 marksStart

The first three terms of a geometric sequence are

2k - 5 \qquad\qquad 4 - k \qquad\qquad k - 6

k

is a constant.
(a) Show that

k

satisfies the equation

k^2 - 9k + 14 = 0

(2 marks)

Given that the sequence converges,
(b) (i) find the value of

k

, giving a reason for your answer,

(ii) find the value of

S_\infty

●●●●●Level 47 marksStart

The first three terms of a geometric series are

k + 2\sqrt{3} \qquad\qquad k \qquad\qquad \sqrt{3}

k

is a constant.

Given that the series converges,
(a) use algebra, making your reasoning clear, to prove that

k = 2\sqrt{3}

(3 marks)
(b) Find the sum to infinity of the series, giving your answer in the form

a\sqrt{3}

a

is an integer.

(2 marks)
(c) Find the smallest value of

n

for which the sum of the first

n

terms of the series exceeds

99\%

of the sum to infinity.

(2 marks)

●●●●●Level 47 marksStart

The first three terms of a geometric series are

\cos\theta \qquad\qquad 1 + \sin\theta \qquad\qquad 4\cos\theta

(a) Show that

5\sin^2\theta + 2\sin\theta - 3 = 0

(3 marks)

Given that

0 < \theta < \dfrac{\pi}{2}

,
(b) find the value of

\theta

, giving your answer to 3 decimal places.

(2 marks)
(c) Find the sum of the first

10

terms of the series.

(3 marks)

●●●●●Level 48 marksStart

An arithmetic sequence has first term

a

and non-zero common difference

d

2

5

11

th terms of the arithmetic sequence form the first three terms of a geometric sequence.
(a) Show that the common ratio of the geometric sequence is

2

. (5 marks)
(b) Given that

a = 4

, find the sum of the first

20

terms of the arithmetic sequence. (2 marks)

●●●●●Level 47 marksStart

A ball is dropped from a height of

2

m onto horizontal ground. After each bounce the ball rises to

65\%

of the height it reached before that bounce.
(a) Find the height the ball reaches after the

5

th bounce, giving your answer to

3

significant figures. (2 marks)
(b) Find the total distance travelled by the ball from the moment it is dropped until the moment it hits the ground for the

6

th time, giving your answer to

3

significant figures. (4 marks)
(c) Assuming the ball continues to bounce indefinitely, find the total distance it travels, giving your answer to

3

significant figures. (3 marks)

●●●●●Level 49 marksStart

A geometric series has first term

12

and sum to infinity

16

.
(a) Show that the common ratio of the series is

\dfrac{1}{4}

. (3 marks)
(b) Find the smallest value of

n

for which the sum to infinity differs from the sum of the first

n

terms by less than

0.001

●●●●●Level 48 marksStart

The first three terms of a geometric progression are

4k

k+3

k

k

is a non-zero constant.
(a) Show that

k^{2} - 2k - 3 = 0

and hence find the two possible values of

k

. (4 marks)
(b) It is given that the common ratio of the progression is positive. Find the sum to infinity. (2 marks)
(c) An arithmetic progression has first term equal to the first term of the geometric progression in part (b), and common difference

5

. Find the least number of terms whose sum exceeds

500

●●●●●Level 49 marksStart

A company makes a profit of £

150\,000

in its first year of trading. In a model, the profit is assumed to grow by

4\%

each year, so the annual profits form a geometric sequence.
(a) Find the profit the model predicts for year

10

. Give your answer to the nearest pound. (2 marks)
(b) Find the total profit the model predicts for the first

15

years of trading. Give your answer to the nearest pound. (3 marks)
(c) Find the first year in which the model predicts an annual profit of more than £

250\,000

. Show your working clearly. (3 marks)

●●●●●Level 48 marksStart

A geometric series has common ratio

r

|r| < 1

. The sum of the first two terms of the series is

9

, and the sum to infinity of the series is

25

.
(a) Find the two possible values of

r

. (4 marks)
(b) For the series with the smaller first term, find the least value of

n

for which the sum of the first

n

24.5

●●●●●Level 46 marksStart

The first, second and fifth terms of an arithmetic sequence, with first term

a

and common difference

d

d \neq 0

, are also the first three terms of a geometric sequence.

The sum of the first

20

terms of the arithmetic sequence is

800

.

Find the smallest value of

n

n

th term of the geometric sequence is greater than one million. (6 marks)

●●●●●Level 46 marksStart

A geometric sequence

u_1, u_2, u_3, \ldots

u_1 = 2

and common ratio

4

.
(a) Show that

\displaystyle\sum_{r=1}^{n} \log_2 u_r = n^2

. (3 marks)
(b) Hence determine the smallest value of

n

u_1 \times u_2 \times u_3 \times \cdots \times u_n > 10^{30}

●●●●●Level 47 marksStart

Sigma notation and recurrence relations

29 questions

LessonNot started

A sequence has terms

u_1, u_2, u_3, \ldots

u_1 = 3

u_{n+1} = \dfrac{u_n + 6}{u_n - 1}

n \geqslant 1

.
(a) Find the values of

u_2

u_3

u_4

.

(2 marks)
(b) Describe the behaviour of the sequence.

(1 mark)
(c) Given that

\displaystyle\sum_{n=1}^{k} u_n = 363

, determine the value of

k

●●●●●Level 46 marksStart

A sequence of numbers

u_1, u_2, u_3, \ldots

u_1 = 5

u_{n+1} = (-1)^{n+1}u_n + k

k

is a constant.
(a) Show that

u_5 = 5

(3 marks)

Given that

\sum_{r=1}^{26} u_r = 88

(b) find the value of

k

.

(2 marks)
(c) Hence find the value of

\sum_{r=1}^{103} u_r

●●●●●Level 47 marksStart

u_1, u_2, u_3, \ldots

u_{n+1} = au_n - b \qquad\qquad u_1 = 4

a

b

are constants.
(a) Find, in terms of

a

b

u_2

u_3

(2 marks)

Given

$\displaystyle\sum_{n=1}^{3} u_n = 36$
$b = 2a + 5$

(b) show that

2a^{2} - 5a - 42 = 0

(3 marks)
(c) Hence find the smaller possible value of

u_4

●●●●●Level 48 marksStart

u_1, u_2, u_3, \ldots

u_1 = 9

u_{n+1} = u_n + 3\cos\left(\frac{n\pi}{2}\right)

(a) Find the value of

u_2

u_3

u_4

and the value of

u_5

(2 marks)
(b) State the order of this periodic sequence.

(1 mark)
(c) Find the value of

\displaystyle\sum_{n=1}^{62} u_n

●●●●●Level 46 marksStart

a_1, a_2, a_3, \ldots

a_1 = k

\qquad a_{n+1} = 1 - \dfrac{1}{a_n}

\quad n \ge 1

k

is a constant such that every term of the sequence is defined.
(a) Show that

a_4 = k

. (4 marks)
(b) Explain why

k

cannot be equal to

1

. (1 mark)
(c) Given that

k = 3

, find the value of

\displaystyle\sum_{r=1}^{200} \left(2a_r - 1\right)

,

giving your answer as an exact fraction. (3 marks)

●●●●●Level 48 marksStart

u_1, u_2, u_3, \ldots

u_1 = 4, \qquad u_{n+1} = \dfrac{u_n - 1}{u_n}, \qquad n \geqslant 1

(a) Show that the sequence is periodic, and state its period. (2 marks)
(b) Find the exact value of

\displaystyle\sum_{r=1}^{200} u_r

●●●●●Level 45 marksStart

A geometric sequence

u_1, u_2, u_3, \ldots

u_1 = 2

and common ratio

4

.
(a) Show that

\displaystyle\sum_{r=1}^{n} \log_2 u_r = n^2

. (3 marks)
(b) Hence determine the smallest value of

n

u_1 \times u_2 \times u_3 \times \cdots \times u_n > 10^{30}

●●●●●Level 47 marksStart

A sequence is defined by

u_1 = 3

u_{n+1} = 2u_n - 1

u_2

u_3

u_4

. (3 marks)
(b) Evaluate

\displaystyle\sum_{n=1}^{4} u_n

●●●●●Level 45 marksStart

A sequence is defined by

u_{1} = 3

u_{n+1} = ku_{n} + 2

k

is a constant.
(a) Find expressions for

u_{2}

u_{3}

k

. (2 marks)
(b) Given that

u_{3} = 10

k

is an integer, find the value of

k

●●●●●Level 45 marksStart

A sequence is defined by

u_{1} = 4

u_{n+1} = \dfrac{2u_{n} - 3}{u_{n} - 1}

.
(a) Show that the sequence is periodic, and state its period. (2 marks)
(b) Write down

u_{99}

. (1 mark)
(c) Find

\displaystyle\sum_{n=1}^{90} u_{n}

●●●●●Level 45 marksStart

Binomial expansion for any index

35 questions

LessonNot started

(a) Find the first three terms in the expansion of

(4 - 3x)^{-\frac{1}{2}}

in ascending powers of

x

.

(4 marks)
(b) State the range of values of

x

for which the expansion in part (a) is valid.

(1 mark)
(c) Find the coefficient of

x^2

in the expansion of

\dfrac{2 + 5x}{\sqrt{4 - 3x}}

●●●●●Level 49 marksStart

(a) Find the first three terms in the expansion of

(4 - 16x)^{\frac{1}{2}}

in ascending powers of

x

.

(4 marks)
(b) State the range of values of

x

for which the expansion in part (a) is valid.

(1 mark)
(c) In the expansion of

(1 + ax)^2(4 - 16x)^{\frac{1}{2}}

the coefficient of

x^2

6

.

Determine the possible values of the constant

a

●●●●●Level 49 marksStart

(a) Find the first three terms in the expansion of

(9 + 2x)^{-\frac{1}{2}}

in ascending powers of

x

.

(4 marks)
(b) State the set of values of

x

for which this expansion is valid.

(1 mark)
(c) When

\dfrac{6 + ax}{\sqrt{9 + 2x}}

is expanded in ascending powers of

x

, the coefficient of

x

is zero.

Determine the value of the constant

a

●●●●●Level 47 marksStart

The diagram shows triangle

ABC

BAC

120°

and the lengths of

AB

AC

(5 + h)

(3 - h)

cm respectively, where

0 < h < 3

.(a) Show that the length of

BC

\sqrt{p}

p = 49 + 2h + h^2

.

(2 marks)
(b) Hence show that, when

h

\sqrt{p} \approx 7 + mh + nh^2

m

n

are rational numbers whose values are to be determined.

(4 marks)

●●●●●Level 46 marksStart

In this question you should assume that $-1 < x < 1$ .
(a) (i) Find the first four terms in the expansion of

(1 + x)^{-2}

in ascending powers of

x

.

(ii) Write down the term in

x^n

in this expansion, where

n

is a positive integer.

(3 marks)
(b) Write down the sum to infinity of the series

1 - x + x^2 - x^3 + \ldots

(1 mark)
(c) Hence or otherwise find and simplify an expression for

2 - 5x + 8x^2 - 11x^3 + \ldots

\dfrac{a + bx}{(1 + x)^2}

a

b

are constants to be determined.

(3 marks)

●●●●●Level 47 marksStart

(a) Find the binomial expansion of

(8 - 3x)^{\frac13}

in ascending powers of

x

, up to and including the term in

x^2

, giving each coefficient in its simplest form. (4 marks)
(b) Use your expansion, with a suitable value of

x

, to find an estimate for

\sqrt[3]{7.7}

, giving your answer to

4

decimal places. (2 marks)

●●●●●Level 46 marksStart

\sqrt{4 - 3x}

in ascending powers of

x

, up to and including the term in

x^{2}

, simplifying the coefficients.

State the set of values of

x

for which the expansion is valid. (5 marks)

●●●●●Level 45 marksStart

Given that the binomial expansion, in ascending powers of

x

(8 + Ax)^{\frac{1}{3}}

is

2 + Bx + Cx^2 + \dfrac{5}{12}x^3 + \ldots

(a) find the values of the constants

A

B

C

.

(5 marks)
(b) State the range of values of

x

for which this expansion is valid.

(1 mark)

For the expansion,
(c) find the coefficient of the term in

x^4

●●●●●Level 48 marksStart

Find the first three terms, in ascending powers of

x

, of the expansion of

(1 + 2x)^{1/2}

, and state the range of validity.

●●●●●Level 43 marksStart

(1 - 3x)^{-2}

up to and including the term in

x^2

, stating the range of validity.

●●●●●Level 43 marksStart

(1 - 2x)^{-2}

in ascending powers of

x

, up to and including the term in

x^{2}

. (3 marks)
(b) Hence find the coefficient of

x^{2}

in the expansion of

\dfrac{2 + 3x}{(1 - 2x)^{2}}

●●●●●Level 45 marksStart

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

in ascending powers of

x

, up to and including the term in

x^{2}

, and state the range of validity. (5 marks)

●●●●●Level 45 marksStart

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

x

, of the binomial expansion of

(1 - x)^{-\frac{1}{2}}

(2 marks)
(b) Hence, for small values of

x

\tan 2x + \frac{3}{\sqrt{\cos 4x}} \approx A + Bx + Cx^2

A

B

C

are constants to be found.

(4 marks)

●●●●●Level 46 marksStart

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.
(a) Find the first three terms, in ascending powers of

x

, of the binomial expansion of

(4 + 3x)^{-\frac{1}{2}}

writing each term in simplest form.

(4 marks)
(b) Using the answer to part (a) and using algebraic integration, estimate the value of

\int_0^{\frac{1}{2}} \frac{2x}{\sqrt{4 + 3x}}\,\mathrm{d}x

giving your answer to 4 significant figures.

(4 marks)

●●●●●Level 48 marksStart