Questions

Q1.

A company that owned a silver mine

 extracted 480 tonnes of silver from the mine in year 1

 extracted 465 tonnes of silver from the mine in year 2
 extracted 450 tonnes of silver from the mine in year 3
and so on, forming an arithmetic sequence.

(a) Find the mass of silver extracted in year 14

(2)
After a total of 7770 tonnes of silver was extracted, the company stopped mining.

Given that this occurred at the end of year N,

(b) show that

N2 − 65N + 1036 = 0
(3)
(c) Hence, state the value of N.
(1)

(Total for question = 6 marks)

Q2.

Ben is saving for the deposit for a house over a period of 60 months.

Ben saves £100 in the first month and in each subsequent month, he saves £5 more than the previous month, so that
he saves £105 in the second month, £110 in the third month, and so on, forming an arithmetic sequence.

(a) Find the amount Ben saves in the 40th month.

(2)
(b) Find the total amount Ben saves over the 60-month period.
(3)
Lina is also saving for a deposit for a house.

Lina saves £600 in the first month and in each subsequent month, she saves £10 less than the previous month, so that
she saves £590 in the second month, £580 in the third month, and so on, forming an arithmetic sequence.

Given that, after n months, Lina will have saved exactly £18 200 for her deposit,

(c) form an equation in n and show that it can be written as

n2 − 121n + 3640 = 0
(3)
(d) Solve the equation in part (c).
(2)
(e) State, with a reason, which of the solutions to the equation in part (c) is not a sensible value for n.
(1)

(Total for question = 11 marks)

Q3.

Given logab = k, find, in simplest form in terms of k,

(i)
(2)

(ii)
(2)

(iii)
(3)

(Total for question = 7 marks)

Q4.

Kim starts working for a company.

 In year 1 her annual salary will be £16 200

 In year 10 her annual salary is predicted to be £31 500
Model A assumes that her annual salary will increase by the same amount each year.

(a) According to model A, determine Kim's annual salary in year 2.

(3)
Model B assumes that her annual salary will increase by the same percentage each year.

(b) According to model B, determine Kim's annual salary in year 2. Give your answer to the nearest £10
(3)
(c) Calculate, according to the two models, the difference between the total amounts that Kim is predicted to earn
from year 1 to year 10 inclusive. Give your answer to the nearest £10
 (3)

(Total for question = 9 marks)

Q5.

A sequence a1, a2, a3, ... is defined by

an + 1 = 4 – an

a1 = 3

Find the value of

(a) (i) a2
(ii) a107
(2)

(b)
(2)

(Total for question = 4 marks)

Q6.

A sequence α1, α2, α3, ... is defined by

where p is a constant.

(a) Find an expression for α2 in terms of p, giving your answer in simplest form.
(1)

Given that an

(b) find the possible values of α2

(6)

(Total for question = 7 marks)

Q7.

In a geometric sequence u1, u2, u3, ...

 the common ratio is r

 u2 + u3 = 6
 u4 = 8
(a) Show that r satisfies

3r2 − 4r − 4 = 0
(3)
Given that the geometric sequence has a sum to infinity,

(b) find u1
(3)
(c) find S∞
(2)

(Total for question = 8 marks)

Q8.

The adult population of a town at the start of 2019 is 25 000

A model predicts that the adult population will increase by 2% each year, so that the number of adults in the
population at the start of each year following 2019 will form a geometric sequence.

(a) Find, according to the model, the adult population of the town at the start of 2032
(3)
It is also modelled that every member of the adult population gives £5 to local charity at the start of each year.

(b) Find, according to these models, the total amount of money that would be given to local charity by the adult
population of the town from 2019 to 2032 inclusive. Give your answer to the nearest £1 000
(3)

(Total for question = 6 marks)

Q9.

The first term of a geometric series is 20 and the common ratio is The sum to infinity of the series is S∞
(a) Find the value of S∞
(2)
The sum to N terms of the series is SN

(b) Find, to 1 decimal place, the value of S12

(2)
(c) Find the smallest value of N, for which S∞ – SN < 0.5
(4)

(Total for question = 8 marks)

Q10.

An arithmetic series has first term a and common difference d.

(a) Prove that the sum of the first n terms of the series is

(4)
A company, which is making 200 mobile phones each week, plans to increase its production.

The number of mobile phones produced is to be increased by 20 each week from 200 in week 1 to 220 in week 2, to
240 in week 3 and so on, until it is producing 600 in week N.

(b) Find the value of N

(2)
The company then plans to continue to make 600 mobile phones each week.

(c) Find the total number of mobile phones that will be made in the first 52 weeks starting from and including week
1.
(5)

(Total for question = 11 marks)

Q11.

(i) An arithmetic series has first term α and common difference d.

Prove that the sum to n terms of this series is

(3)
(ii) A sequence u1, u2, u3,... is given by
 un = 5n + 3(–1)n
Find the value of
(a) u5
(1)

(b)
(3)

(Total for question = 7 marks)

Q12.

In this question you must show detailed reasoning.

Owen wants to train for 12 weeks in preparation for running a marathon.

During the 12-week period he will run every Sunday and every Wednesday.

 On Sunday in week 1 he will run 15 km

 On Sunday in week 12 he will run 37 km
He considers two different 12-week training plans.

In training plan A, he will increase the distance he runs each Sunday by the same amount.

(a) Calculate the distance he will run on Sunday in week 5 under training plan A.
(3)
In training plan B, he will increase the distance he runs each Sunday by the same percentage.

(b) Calculate the distance he will run on Sunday in week 5 under training plan B. Give your answer in km to one
decimal place.
(3)
Owen will also run a fixed distance, x km, each Wednesday over the 12-week period.

Given that

 x is an integer
 the total distance that Owen will run on Sundays and Wednesdays over the 12 weeks will not exceed 360
km
(c) (i) find the maximum value of x, if he uses training plan A,
(ii) find the maximum value of x, if he uses training plan B.
(5)

(Total for question = 11 marks)

Q13.
A geometric series has first term a and common ratio r.

(a) Prove that the sum of the first n terms of this series is given by

(3)

The second term of a geometric series is −320 and the fifth term is

(b) Find the value of the common ratio.

(2)
(c) Hence find the sum of the first 13 terms of the series, giving your answer to 2 decimal places.
(3)

(Total for question = 8 marks)

Q14.

Adina is saving money to buy a new computer. She saves £5 in week 1, £5.25 in week 2, £5.50 in week 3 and so
on until she
has enough money, in total, to buy the computer.

She decides to model her savings using either an arithmetic series or a geometric series.

Using the information given,

(a) (i) state with a reason whether an arithmetic series or a geometric series should be used,
(ii) write down an expression, in terms of n, for the amount, in pounds (£), saved in week n.
(3)
Given that the computer Adina wants to buy costs £350

(b) find the number of weeks it will take for Adina to save enough money to buy the computer.
(4)

(Total for question = 7 marks)

Q15.

A colony of bees is being studied.

The number of bees in the colony at the start of the study was 30 000
Three years after the start of the study, the number of bees in the colony is 34 000

A model predicts that the number of bees in the colony will increase by p % each year, so that the number of bees in
the colony at the end of each year of study forms a geometric sequence.

Assuming the model,

(a) find the value of p, giving your answer to 2 decimal places.

(3)
According to the model, at the end of N years of study the number of bees in the colony exceeds 75 000

(b) Find, showing all steps in your working, the smallest integer value of N.
(5)

(Total for question = 8 marks)

Q16.

A metal post is repeatedly hit in order to drive it into the ground.

Given that

 on the 1st hit, the post is driven 100 mm into the ground
 on the 2nd hit, the post is driven an additional 98 mm into the ground
 on the 3rd hit, the post is driven an additional 96 mm into the ground
 the additional distances the post travels on each subsequent hit form an arithmetic sequence
(a) show that the post is driven an additional 62 mm into the ground with the 20th hit.
(1)
(b) Find the total distance that the post has been driven into the ground after 20 hits.
(2)
Given that for each subsequent hit after the 20th hit

 the additional distances the post travels form a geometric sequence with common ratio r
 on the 22nd hit, the post is driven an additional 60 mm into the ground
(c) find the value of r, giving your answer to 3 decimal places.
(2)
After a total of N hits, the post will have been driven more than 3 m into the ground.

(d) Find, showing all steps in your working, the smallest possible value of N.
(4)

(Total for question = 9 marks)

Q17.
(i) A geometric sequence has first term 4 and common ratio 6
Given that the nth term is greater than 10100, find the minimum possible value of n.
(3)
(ii) A different geometric sequence has first term a and common ratio r.
Given that

 the second term of the sequence is −6

 the sum to infinity of the series is 25
(a) show that
25r2 − 25r − 6 = 0
(3)
(b) Write down the solutions of
25r2 − 25r − 6 = 0
(1)
Hence,
(c) state the value of r, giving a reason for your answer,
(1)
(d) find the sum of the first 4 terms of the series.
(2)

(Total for question = 10 marks)

Q18.

Solutions based entirely on graphical or numerical methods are not acceptable in this question.

(i) Solve, for 0 ≤ θ <180°, the equation

3 sin (2θ − 10°) = 1

giving your answers to one decimal place.
(4)
(ii) The first three terms of an arithmetic sequence are

where α is a constant.
(a) Show that 2 cos α = 3 sin2α
(3)
Given that π < α < 2π,
(b) find, showing all working, the value of α to 3 decimal places.
(5)

(Total for question = 12 marks)

Mark Scheme
Q1.

Q2.
Q3.
Q4.
Q5.
Q6.
Q7.
Q8.
Q9.
Q10.
Q11.
Q12.
Q13.
Q14.
Q15.
Q16.
Q17.
Q18.