Modelling with Sequences & Series (Edexcel A Level Maths: Pure): Exam Questions

A ball is dropped from the top of a building and is allowed to bounce until it comes to rest.

The height the ball reaches after each bounce is modelled by the geometric sequence

where is the height, in metres, of the ^th bounce.

(i) Find the height the ball reaches after its first bounce.

(ii) Find the height the ball reaches after its fifth bounce.

State whether the sequence is increasing or decreasing.

(i) Find the sum of the heights that the ball reaches after 10 bounces.

(ii) Explain how to use your answer in part (c)(i) to find the total distance travelled by the ball from the moment it first hits the ground to the moment it returns to the ground after the 10^th bounce.

A circular training track for cyclists is made up of several lanes.

• The inner lane has a radius of 10 metres.

• The radius of the next lane outwards is always 4 metres greater than the radius of the previous lane.

You may assume that the lap distance of each lane is the circumference of the circle with the radius indicated above.

Show that the lap distances of each lane form an increasing arithmetic sequence and state the exact values of the first term and common difference.

During a training session, a cyclist is expected to complete two laps of each lane, starting with the inner lane before moving on to the next one.

Find, to the nearest metre, the distance completed by a cyclist during a training session that uses the first 6 lanes.

A different cyclist completes a training session that uses the first 10 lanes.

After that, they continue cycling around the outer lane.

Find the minimum number of laps of the outer lane that this cyclist must do in order to have travelled a total distance of at least 6000 metres.

The original value of a new car is £30 000.

The value of the car after years is £.

Model 1 is the arithmetic sequence

Model 2 is the geometric sequence

(i) Find, in years, the time predicted by model 1 for the car's value to drop to half of its original value.

(ii) Find, in years to 3 significant figures, the time predicted by model 2 for the car's value to drop to half of its original value.

Determine which model predicts a greater value of the car after 5 years.

(i) State a limitation of model 1.

(ii) The value predicted by model 2 never reaches £0. Explain why this may be realistic.

A tree farmer plants trees at the start of each year then sells those trees at the end of that year.

The next year, the tree farmer plants twice as many trees as the previous year, plus one additional tree.

For example

• The tree farmer plants 7 trees at the start of year 1

• The tree farmer sells those 7 trees at the end of year 1

• The tree farmer plants  trees at the start of year 2

• The tree farmer sells those 15 trees at the end of year 2, and so on

If the tree farmer plants 9 trees in year 1, find the number of trees planted in years 2, 3 and 4.

At the very start of which year will the tree farmer be able to claim that they have sold over 600 trees in total?

A competitor is running a kilometre race.

She runs each of the first kilometres at a steady pace of minutes per kilometre.

After the first kilometres, she begins to slow down.

In order to estimate her finishing time, the time that she will take to complete each subsequent kilometre is modelled to be greater than the time that she took to complete the previous kilometre.

Using the model, show that her time to run the first kilometres is estimated to be minutes seconds.

Using the model, show that her estimated time, in minutes, to run the th kilometre, for , is

Using the model, estimate the total time, in minutes and seconds, that she will take to complete the race.

June is training to run a marathon.

• For the first week of training, she runs 3 miles

• Each subsequent week, she increases the number of miles by  miles

• June trains for  weeks, where

Find an expression in terms of  for the number of miles June runs in her 10^th week of training.

Find an expression in terms of  and  for the total distance June runs during the training period.

You are given that

• June runs twice as far in week 4 than she does in week 2,

• June runs a total distance of 570 miles during the training period.

Find the values of  and .

Stephen opens a savings account and pays in an initial amount of £600.

• At the end of each year, compound interest of 1.2% is added to whatever amount of money is in the savings account

• At the start of each new year, Stephen pays another £600 into his account

Show that, at the end of two years, Stephen has £1221.69 in the account.

Show that, by the end of year , Stephen has

pounds in the account.

Write the total amount (in pounds) in Stephen’s account after  years in the form

where and are constants to be found.

After which year will the total amount in Stephen's account first exceed £10,000?

State one assumption that has been made when using the model above.

A healthy unicorn will breathe out one million particles of magic dust with every breath.

• However, as a unicorn dies, the number of particles of magic dust reduces by 20% with each breath

• A unicorn’s last breath is the first breath to contain under 100 particles of magic dust

• You may assuming that a unicorn takes a breath every 10 seconds

Find the time it takes for the unicorn to die, from the moment it takes its last healthy breath.

Including its last healthy breath, find to the nearest whole number the total number of magic dust particles a unicorn breathes out as it dies.

Show that even if it continued to live and breathe with under 100 particles of magic dust in each breath, a unicorn will never be able to breathe out more than 5 million particles of magic dust when dying (including its last healthy breath).

A model maker cuts an original tube into smaller tubes, as shown below.

• The length of the tallest of the smaller tubes is cm

• Each smaller tube is a tenth shorter than the one on its left

Show that no matter how many smaller tubes the model maker uses, the original tube that they are cut from need not be any longer than cm.

An original tube of length cm is cut into smaller tubes, which are then stacked into a tower.

• The tallest of the smaller tubes is cm and is at the bottom of the tower.

• The next smaller tube is always 1 cm shorter than the previous smaller tube.

If there are  tubes in the tower, show that

A ball is dropped from a height of m.

• It bounces to a height of m

• The height of each subsequent bounce is 25% shorter than the height of the previous bounce

Show that no matter how many times the ball bounces, it will not travel further than a total distance of m.

State one assumption that has been made using this model when calculating the total distance in part (a).

A florist grows rose bushes. 

• At the start of each year, the florist will dig up all their rose bushes, cutting flowers that are of high enough quality to sell in their shop and donating what’s left to local garden projects

• Each year the soil quality improves such that the florist can grow two more than four times as many rose bushes as in the previous year

• For example, if the florist grew three rose bushes one year then the following year they would grow  rose bushes

If the florist grows  rose bushes in year 1 and grows 362 rose bushes in year 4, find the value of .

On average, a rose bush will create 6 roses of high enough quality to be able to be sold at the florists’ shop. 

Determine in which year the florist will have grown enough rose bushes to supply over 5000 roses to their shop that year.

Mary wishes to open a savings account and is comparing two options.

In the first option, Mary puts an initial amount of £4000 into an account then

• at the end of each year, compound interest of 0.85% is added to whatever amount of money is in the account

• at the start of each new year, Mary pays another £4000 into the account

In the second option, Mary puts an initial amount of £3000 into an account then

• at the end of each year, compound interest of 1.1% is added to whatever amount of money is in the account

• at the start of each new year, Mary pays another £3000 into the account

If Mary wishes to reach £40,000 as soon as possible, which account should she use and how long will it take (to the nearest whole year)?

A lane on a training track for cyclists is in the shape of a rectangle and two semi-circles, as shown below. 

The track has 12 lanes in total, where

• the shortest, inner lane, as shown in the diagram, has straight runs of m, and semi-circles at each end with a diameter of m

• each lane moving outwards increases the diameter by m compared to the previous lane

Describe the sequence of the total lengths of each lane in order from the smallest to largest.

The total distance covered by the lengths of all 12 lanes is m.

Given that  and  are integers, find the value of and show that

where is an integer to be found.

It is recommended that lanes are at least 2 m wide to allow sufficient space between cyclists in different lanes.

Find the least value of  and the corresponding value of .