CONFLICTOS TERRITORIALES
B) LA CARGA DE LA PRUEBA
Here are two different sequences.
a 1, 4, 9, ...
b 10, 4, 1.6...
Decide whether each sequence is geometric and fi nd the common ratio if there is one.
Solution
a Work out the ratios between the terms.
4 ÷ 1 = 1 9 ÷ 4 = 2.25
There is no common ratio between successive terms so the sequence is not a geometric progression.
b Work out the ratios between the terms.
4 ÷ 10 = 0.4 1.6 ÷ 4 = 0.4
These are the same so the sequence is a geometric progression.
The common ratio,
r
, is 0.4.10 × r = 4 so r = 4 ÷ 10 4 × r = 1.6 so r = 1.6 ÷ 4
Check: 10 × 0.4 = 4 ✓ and 4 × 0.4 = 1.6 ✓
Practising skills
1 Decide whether each sequence is a geometric progression.
a 1, 2, 3, 4…
b 1, 2, 4, 8…
c 1, 2, 4, 7…
d 80, 40, 20, 10…
2 A geometric progression has fi rst term 3 and common ratio 3.
a Write down the fi rst four terms of the sequence.
b Describe these numbers.
3 The sequences are geometric progressions.
Write down the common ratio for each one.
a 1, 4, 16, 64…
b 4, 12, 36, 108…
c 4, 2, 1, 12
d 10, 1, 0.1, 0.01…
4 The sequences are geometric progressions.
Write down the next two terms for each one.
a 2, 10, 50, 250…
b 50, 5, 0.5, 0.05…
c 8, 12, 18, 27…
d 6, 1.2, 0.24, 0.048…
5 The second and third terms of a geometric progression are 6 and 18.
Work out
a the common ratio b the fi rst term c the fi fth term.
Developing fl uency
1 Here are the fi rst three terms of some sequences.
a Decide whether the numbers below form geometric progressions.
i 200, 300, 450, … ii 150, 120, 100, … iii 24, 30, 37.5, … In each case
b Find the ratios between i the fi rst two terms.
ii the second and third terms.
Unit 6 Geometric progressions Band g 2 In each case the fi rst and third terms of a geometric progression are given. You are asked to
work out one of the other terms.
a 1 and 36. Work out the second term.
b 4 and 1. Work out the second term.
c 2 and 12.5. Work out the fourth term.
3 The fi rst term of a sequence is 1 and the fi fth term is 81.
Write down the second, third and fourth terms of the sequence if it is a a geometric progression
b linear sequence.
4 £5000 is invested at a compound interest rate of 2%.
a Work out how much it is worth after 1 year.
b Work out how much it is worth after 2 years.
c Work out how much it is worth after 3 years.
d The amounts form a geometric progression. What is its common ratio?
5 The half-life of fermium-253 is 3 days.
Half of the remaining radioactive atoms in a sample of fermium-253 will decay over a period of 3 days.
How many days is it before only 6.25% of the original radioactive atoms remain?
6 Elsa has a rich but eccentric uncle. She receives the following email from him at the start of the summer.
Dear Elsa,
I would like to give you a daily allowance for your summer holidays.
Which of these would you prefer?
£100 per day,
or, £10 on day one, £20 on day 2, £30 on day 3 and so on.
or, 1p on day 1, 2p on day 2, 4p on day 3 and so on.
See you soon!
Uncle Jim
a Work out how much would Elsa receive under each option on i day 7
ii day 14 iii day 28.
b i Which option gives Elsa the most money?
ii Work out the total amount Elsa receives under this option after 4 weeks.
(Hint: Find how much Elsa receives after 1 day, 2 days, 3 days and so on. What is special about these numbers?)
Exam-styleExam-style
Problem solving
1 Here is part of Pascal’s triangle.
a The totals of the fi rst three rows are 1, 2, 4.
What are the next three terms of this sequence?
b What is the term-to-term rule for this sequence?
c What is the position-to-term rule?
d Which row has a total of 1024?
2 Jason bought a new car for £10 000.
Its value depreciated by 10% each year.
So its value £ V, after
n
years was given by V = 10 000 × 0.9na Find the value of the car after
i 1 year ii 2 years iii 3 years.
b The answers to part a give a sequence of values of the car.
What is the term-to-term rule for this sequence?
c After how many years is the value of the car fi rst less than £3000?
3 Payday loan companies offer loans for a short period of time (typically for a few weeks).
A payday loan company charges interest at 30% per month. To work out how much you owe you need to multiply the previous month’s amount by 1.3.
a If you borrow £100, how much will you owe them after 2 months?
b The companies are not supposed to lend money to people for a long period of time.
How much would you owe if you borrowed £100 and did not pay it back for a year?
c The amount of money in the world is about three trillion pounds (£3 000 000 000 000). If you borrow £100, roughly how long would it take before you needed to pay back more money than there is in the whole world?
Exam-style
1 For these geometric progressions a Find the common ratio.
b Write down the next two terms in the progressions below.
i 3, 6, 12, 24, … ii 64, 16, 4, 1, …
2 In each case the fi rst and third terms of a geometric progression are given. You are asked to work out two other terms.
a 3 and 12. Work out the fi fth and tenth terms.
b 5 and 0.2. Work out the second and fourth terms.
Strand 3 Functions and graphs
Unit 1 Band f
Real-life graphs
Foundation 1
Unit 5 Band h
Finding equations of straight lines
Page 115
Unit 4 Band g
Plotting quadratic and cubic graphs Foundation 1
Unit 8 Band i
Perpendicular lines
Higher 2
Unit 9 Band j
Inverse and composite functions
Higher
Unit 10 Band j Exponential functions
Higher 2 Unit 11 Band k
Trigonometrical functions
Higher 2 Unit 12 Band k
The equation of a circle
Higher 2
Unit 2 Band f
Plotting graphs of linear functions
Foundation 1
Unit 6 Band h
Quadratic functions
Page 122
Unit 3 Band g
The equation of a straight line Foundation 1
Unit 7 Band h
Polynomial and reciprocal functions
Page 128
Units 1–4 are assumed knowledge for this book. They are reviewed and extended in the Moving on section on page 114.