AS Core Maths - TAM Online Session 5: Sequences & Series

A warm-up question before we get started…

A RI THM ETIC PROGR ESSIONG EO METR ICSERIES

51 is the 6th pentagonal number.

What is the 100th pentagonal number?

Session content

Sequences

Recurrence relations

Sigma notation

Arithmetic Progressions

Geometric Progressions

NB: This is content based on chapter 7 in the AS Core textbook

Sequences

Recurrence Relations

Sigma Notation

Arithmetic Sequence

1 2 3 1

100

1, 4, 7, ... 3

?n nu u u u u

u

3 5 7 9 11 13

Arithmetic Series

Arithmetic Series

6 3 133 5 7 9 11 13 48

21 4 7 ... 298

Arithmetic Progressions & Series

1 2 3 1

100

100

1001

1, 4, 7, ... 3

1 99 3 298

n n

rr

u u u u u

u

S u

APs: the formulae

1 2 3

1

, , 2 , ...

n

n

n rr

u a u a d u a d

u

S u

Typical APs Exam Question (Core 2 Jan 07 - Q8 [5 marks])The 7th term of an arithmetic progression is 6.

The sum of the first 10 terms of the progression is 30.

Find the 5th term of the progression.

Geometric Progressions & Series

1 2 3 4 1

10

5

51

2, 6, 18, 54, ... 3n n

rr

u u u u u u

u

S u

GPs: the formulae2 3

1 2 3 4

1

, , , , ...

n

n

n rr

u a u ar u ar u ar

u

S u

1 17 8 10 11

2 2

2 5 8 11

1 1.1 1.2 1.3

1 1.1 1.11 1.111

1 5 7 3

2 6 6 2

2 2 2 21 2 3 4

2 3 4n n n n 1 1 1

12 3 4

1 1 1 1

2 3 4a a a a

1 1.1 1.21 1.331

1 1 1 1

2 6 12 36

2 4 8 16

3 927 162

4 2

AP or GP?

Typical GPs Exam Question

Find two different geometric progressions with second term 18 and fourth term 2.

Find the sum of the first 100 terms in each case.

Sum to infinity of a geometric series2 3 10

2 3 100

2 3

1 1 1 1...

4 4 4 4

1 1 1 1...

4 4 4 4

1 1 1...

4 4 4

2 3 41 1 1 1 1

...4 4 4 4 3

Infinite GPs: the formula

2 3 11

...1

n

na r

a ar ar ar arr

n

Typical Infinite GPs Exam Question(Core 2 Jan 08 - Q8 [5 marks]) The second term of a geometric progression is 18 and the fourth term is 2. The common ratio is positive. Find the sum to infinity of this progression.

Session content check

Sequences

Recurrence relations

Sigma notation

Arithmetic Progressions

Geometric Progressions

AQA Core 2 Jun 12 – Q1 [5 marks]

MEI Core 2 Jan 09 - Q6 [5 marks]

OCR Core 2 Jun 12 – Q5 [8 marks]

MEI Core 2 Jun 08 - Q2 [3 marks]

MEI Core 2 Jun 08 - Q8 [5 marks]The 11th term of an arithmetic progression is 1. The sum of the first ten terms is 120. Find the 4th term.

MEI Core 2 Jun 09 - Q11(i) [5 marks]

MEI Core 2 Jun 09 - Q11(ii) [7 marks]