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P2 Chapter 5 :: Sequences & Series

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Chapter Overview

 

1:: Arithmetic Series

The first term of a geometric sequence is 3 and the second term 1. Find the sum to infinity.

2:: Geometric Series

 

3:: Sigma Notation

 

4:: Recurrence Relations

NEW TO A LEVEL 2017!

Identifying whether a sequence is increasing, decreasing, or periodic.

Understanding the real world problems

4:: Modeling with Series

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Types of sequences

2, 5, 8, 11, 14, …

+3

+3

+3

This is a:

Arithmetic Sequence

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Geometric Sequence

(We will explore these later in the chapter)

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3, 6, 12, 24, 48, …

 

 

 

 

 

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1, 1, 2, 3, 5, 8, …

This is the Fibonacci Sequence. The terms follow a recurrence relation because each term can be generated using the previous ones.

We will encounter recurrence relations later in the chapter.

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🖉 An arithmetic sequence is one which has a common difference between terms.

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The fundamentals of sequences

 

 

 

The position of the term in the sequence.

 

 

 

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1st Term

2nd Term

3rd Term

 

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Example 1

 

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Example 2

 

 

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Further Examples

 

 

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Test Your Understanding

Edexcel C1 May 2014(R) Q10

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Exercise 5A

Pearson Pure Mathematics 2

Pages 82-83

Extension

 

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Series

A series is a sum of terms in a sequence.

You will encounter ‘series’ in many places in A Level:

 

 

 

 

 

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Arithmetic Series

 

 

 

 

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Let’s prove it!

 

Example:

 

Proving more generally:

The idea is that each pair of terms, at symmetrically opposite ends, adds to the same number.

Fro Exam Note: The proof has been an exam question before. It’s also a university interview favourite!

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Alternative Formula

 

 

 

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Examples

Find the sum of the first 30 terms of the following arithmetic sequences…

1

 

2

3

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Edexcel C1 Jan 2012 Q9

 

 

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Test Your Understanding

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Exercise 5B

Pearson Pure Mathematics 2

Pages 85-86

 

Extension

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3

 

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Solution to Extension Q2

 

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Recap of Arithmetic vs Geometric Sequences

2, 5, 8, 11, 14, …

+3

+3

+3

This is a:

Arithmetic Sequence

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Geometric Sequence

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3, 6, 12, 24, 48, …

 

 

 

 

 

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🖉 A geometric sequence is one in which there is a common ratio between terms.

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Quickfire Common Ratio

 

 

 

1

 

 

2

 

 

3

 

 

4

 

 

5

 

 

6

 

 

7

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An alternating sequence is one which oscillates between positive and negative.

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Arithmetic Sequence

Geometric Sequence

 

 

 

3, 6, 12, 24, …

40, -20, 10, -5, …

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Further Example

[Textbook] The second term of a geometric sequence is 4 and the 4th term is 8. The common ratio is positive. Find the exact values of:

  1. The common ratio.
  2. The first term.
  3. The 10th term.

 

 

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Further Example

 

 

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Exam Note: This kind of question has appeared in the exam multiple times.

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Test Your Understanding

All the terms in a geometric sequence are positive.

The third term of the sequence is 20 and the fifth term 80. What is the 20th term?

 

 

 

 

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Exercise 5C

Pearson Pure Mathematics 2

Pages 90-91

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Arithmetic Series

Geometric Series

 

 

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Proof:

 

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Exam Note: This once came up in an exam. And again is a university interview favourite!

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Examples

 

 

Find the sum of the first 10 terms.

 

 

 

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Geometric Series

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Harder Example

 

 

 

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Test Your Understanding

Edexcel C2 June 2011 Q6

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Exercise 5D

Pearson Pure Mathematics 2

Pages 93-94

Extension

 

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Divergent vs Convergent

1 + 2 + 4 + 8 + 16 + ...

What can you say about the sum of each series up to infinity?

1 – 2 + 3 – 4 + 5 – 6 + …

1 + 0.5 + 0.25 + 0.125 + ...

 

This is divergent – the sum of the values tends towards infinity.

This is divergent – the running total alternates either side of 0, but gradually gets further away from 0.

This is convergent – the sum of the values tends towards a fixed value, in this case 2.

This is divergent . This is known as the Harmonic Series

 

 

Definitely NOT in the A Level syllabus, and just for fun...

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Sum to Infinity

1 + 0.5 + 0.25 + 0.125 + ...

1 + 2 + 4 + 8 + 16 + ...

Why did this infinite sum converge (to 2)…

…but this diverge to infinity?

 

 

 

 

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Quickfire Examples

 

 

 

 

 

 

 

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Further Examples

[Textbook] The fourth term of a geometric series is 1.08 and the seventh term is 0.23328.

  1. Show that this series is convergent.
  2. Find the sum to infinity of this series.

 

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Test Your Understanding

Edexcel C2 May 2011 Q6

 

 

 

 

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Exercise 5E

Pearson Pure Mathematics 2

Pages 96-97

Extension

 

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2

 

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Sigma Notation

 

What does each bit of this expression mean?

The Greek letter, capital sigma, means ‘sum’.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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Determining the value

 

 

First few terms?

 

 

 

Final result?

 

 

 

Be careful, there are 11 numbers between 5 and 15 inclusive. Subtract and +1.

 

 

 

 

 

 

 

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Solomon Paper A

Testing Your Understanding

 

 

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On your calculator

 

 

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Exercise 5F

Pearson Pure Mathematics 2

Pages 98-99

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Recurrence Relations

 

 

 

 

This is known as a term-to-term sequence, or more formally as a recurrence relation, as the sequence ‘recursively’ refers to itself.

We need the first term because the recurrence relation alone is not enough to know what number the sequence starts at.

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Example

 

Edexcel C1 May 2013 (R)

 

 

 

 

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Test Your Understanding

Edexcel C1 Jan 2012

 

 

 

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Exercise 5G

Pearson Pure Mathematics 2

Pages 101-102

 

 

 

1

2

3

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Solution to Extension Question 3

 

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A somewhat esoteric Futurama joke explained

 

 

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Increasing, decreasing and periodic sequences

 

 

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Textbook Error: It uses the term ‘increasing’ when it means ‘strictly increasing’.

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Exercise 5H

Pearson Pure Mathematics 2

Pages 103-104

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Modelling

 

[Textbook] Bruce starts a new company. In year 1 his profits will be £20 000. He predicts his profits to increase by £5000 each year, so that his profits in year 2 are modelled to be £25 000, in year 3, £30 000 and so on. He predicts this will continue until he reaches annual profits of £100 000. He then models his annual profits to remain at £100 000.

  1. Calculate the profits for Bruce’s business in the first 20 years.
  2. State one reason why this may not be a suitable model.
  3. Bruce’s financial advisor says the yearly profits are likely to increase by 5% per annum. Using this model, calculate the profits for Bruce’s business in the first 20 years.

 

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Geometric Modelling Example

[Textbook] A piece of A4 paper is folded in half repeatedly. The thickness of the A4 paper is 0.5 mm.

  1. Work out the thickness of the paper after four folds.
  2. Work out the thickness of the paper after 20 folds.
  3. State one reason why this might be an unrealistic model.

 

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b

c

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Test Your Understanding

Edexcel C2 Jan 2013 Q3

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Exercise 5I

Pearson Pure Mathematics 2

Pages 105-106

[AEA 2007 Q5]

 

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Extension

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