A-Level Maths (Edexcel 9MA0) · Pure Year 2 · Chapter 3: Sequences and series · Lesson 5

Modelling with series

114 marks

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Quick questions to start. They use what you need today.

A1
Needed today
$£2000$ is invested at $3%$ a year compound interest. Find its value after $5$ years.
[2 marks]
Answer
A2
Needed today
Find the least integer $n$ for which $1.0 4^{n} > 2$ .
[2 marks]
Answer
A3
Common slip
Ana earns $£24 000$ in year $1$ . Her salary rises by $£800$ each year. Write her salary in year $n$ .
[2 marks]
Answer
A4
From chapter 3
A car worth $£12 000$ loses $15%$ of its value each year. Write its value after $n$ years.
[2 marks]
Answer

Example, then your turn

A-Level

Your teacher works through each example with you. Then try the one beside it.

Example 1

Sam saves $£50$ in month $1$ , and $5$ more each month than the month before. How much does he save in $24$ months?

Hint The amounts go up by the same amount each month, so this is an arithmetic series with

a = 50

d = 5

B1
Mia walks $30$ minutes in week $1$ and $4$ minutes more each week. Find the total time she walks in the first $20$ weeks, and the first week in which she walks more than $100$ minutes.
[5 marks]
Answer

Example 2

$£5000$ is invested at $4%$ a year. After how many whole years is it first worth more than $£8000$ ?

Hint Write

5000 \times 1.0 4^{n} > 8000

, divide by

5000

, then take logs.

B2
A van worth $£18 000$ loses $12%$ of its value each year. After how many whole years is it first worth less than $£5000$ ?
[4 marks]
Answer

Example 3

Raj pays $£1000$ into an account at the start of each year. It earns $3%$ a year. Find the value at the end of year $10$ .

Hint At the end of year

10

the first

£1000

has grown for

10

years and the last for

1

year. Add the geometric series with first term

1000 \times 1.03

and

10

terms.

B3
Lily pays $£500$ into an account at the start of each year. It earns $5%$ a year. Find the value at the end of year $8$ .
[5 marks]
Answer

Example 4

Job A pays $£30 000$ in year $1$ and rises by $£1500$ a year. Job B pays $£30 000$ in year $1$ and rises by $4%$ a year. Which pays more in total over $10$ years?

Hint Job A is arithmetic and Job B is geometric. Find each sum to

10

terms.

B4
Pay scale P starts at $£20 000$ and rises by $£1200$ a year. Pay scale Q starts at $£20 000$ and rises by $5%$ a year. Find the first year in which Q pays more than P that year, and the total each pays over the first $10$ years.
[5 marks]
Answer

Example 5

A ball is dropped from $10$ m. Each bounce reaches $0.6$ of the height before. Find the total distance it travels.

Hint It falls

10

m once. After that each height is travelled twice, up and down:

2 (6 + 3.6 + \dots)

B5
A ball is dropped from $8$ m. Each bounce reaches $0.7$ of the height before. Find the total distance it travels, and give one criticism of this model.
[5 marks]
Answer

Practice

A-Level

C1
A shop sells $400$ units in month $1$ , and $25$ more each month than the month before. Find the total sold in the first $12$ months, and the least number of months for the total to pass $30000$ .
[5 marks]
Answer

C2
$£3000$ is invested at $2.5%$ a year compound interest. Find its value after $5$ years, and the least number of whole years for it to be worth more than $£4000$ .
[5 marks]
Answer
C3
A factory makes $2000$ tonnes in year $1$ . Each year it makes $6%$ less than the year before. Find the total made in the first $15$ years. Show that the total made can never be more than $33334$ tonnes.
[5 marks]
Answer
C4
A population of $400$ birds grows by $20%$ a year. After how many whole years is it first more than $2000$ ? Give one reason the model may not hold for many years.
[4 marks]
Answer

Exam-style questions

Edexcel exam style

D1
Priya runs $5$ km on day $1$ . Each day she runs $0.4$ km more than the day before, until she reaches $21$ km a day.
(a)
How far does she run on day $15$ ?
[2 marks]
Answer
(b)
Find the total distance she runs in the first $15$ days.
[2 marks]
Answer
(c)
On which day does she first run $21$ km?
[2 marks]
Answer
(d)
After that she runs $21$ km every day. Find the total distance she runs in the first $60$ days.
[3 marks]
Answer
Total for D1: 9 marks

D2
Lena says: "In month $30$ I save $£390$ , so in the first $30$ months I save $30 \times 390 = £11 700$ ."
Lena is wrong.
(a)
Lena saves $£100$ in month $1$ and $£10$ more each month. Find the correct total for $30$ months, and explain her mistake.
[4 marks]
Answer
(b)
Tom saves $£300$ in month $1$ and $£5$ more each month. Find the least number of months after which Lena's total is more than Tom's.
[5 marks]
Answer
Total for D2: 9 marks
D3
Town A has $25000$ people and its population falls by $2%$ a year. Town B has $18000$ people and its population rises by $3%$ a year. After how many whole years does Town B first have more people than Town A?
[6 marks]
Answer
D4
A ball is dropped from $2.5$ m. After each bounce it reaches $0.8$ of the height before.
(a)
Find the height it reaches after the third bounce.
[2 marks]
Answer
(b)
Find the distance it has travelled when it hits the ground for the tenth time.
[4 marks]
Answer
(c)
In a better model the ball stops once a bounce would reach less than $0.05$ m. How many bounces does it make?
[3 marks]
Answer
(d)
Find the total distance the ball travels in this model, and compare it with the total if it bounced for ever.
[3 marks]
Answer
Total for D4: 12 marks
D5
Omar pays $£2000$ into an account at the start of each year. It earns $4%$ a year, added at the end of each year.
(a)
Show that the value at the end of year $n$ is $£52 000 (1.0 4^{n} - 1)$ .
[3 marks]
(b)
Find the least $n$ for which the value is more than $£50 000$ .
[3 marks]
Answer
(c)
His sister puts one lump sum into the same account now and pays nothing more. Find how much she must pay in to have the same amount as Omar at the end of year $15$ .
[3 marks]
Answer
Total for D5: 9 marks
D6
A mine produces $8000$ tonnes in year $1$ . Each year it produces $5%$ less than the year before. It closes after the last year in which it produces at least $3000$ tonnes.
(a)
Find how many years the mine is open.
[4 marks]
Answer
(b)
Find the total produced while it is open.
[3 marks]
Answer
(c)
Find the total the model says the mine could ever produce, and what fraction of it is produced while it is open.
[2 marks]
Answer
Total for D6: 9 marks

Extension

Stretch

No route is given. The hints are at the end of the sheet. Use one at a time.

E1
A loan of $£10 000$ is taken out. At the end of each month $1%$ interest is added, then a repayment of $£ R$ is made. The loan is paid off after exactly $24$ repayments. Find $R$ .
[9 marks]
Hint 1 · What to try
Write the amount owed after $1$ , $2$ and $3$ months. Look for a geometric series in $R$ .
Hint 2 · The first line
After $n$ months the amount owed is $10000 \times 1.0 1^{n} - R (1 + 1.01 + \dots + 1.0 1^{n - 1})$ .
Hint 3 · The full method
Set the amount owed after $24$ months to $0$ and sum the series.
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can decide whether a situation is arithmetic or geometric and write its terms and sums.
I can find the least number of years or months for a total or a term to pass a value, using logs where needed.
I can model money paid in at the start of each year, or a loan paid off each month.
I can criticise a model and say when it stops being realistic.

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