Transcript MEI PowerPoint Template

AS Core Maths - TAM Online
Session 5: Sequences & Series
A warm-up question before we get started…
A RI THM ETIC PROGR ESSION
G 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
45
21
-80
24
Arithmetic Sequence
u1  1,
u100  ?
u2  4,
u3  7,
...
un 1  un  3
Arithmetic Series
3  5  7  9  11  13 
Arithmetic Series
6   3  13
3  5  7  9  11  13 
 48
2
1  4  7  ...  298 
Arithmetic Progressions & Series
u1  1, u2  4, u3  7, ...
u100  1  99  3  298
100
S100   ur 
r 1
un 1  un  3
APs: the formulae
u1  a, u2  a  d , u3  a  2d , ...
un 
n
S n   ur 
r 1
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
u1  2, u2  6, u3  18, u4  54, ... un 1  3un
u10 
5
S5   ur 
r 1
GPs: the formulae
u1  a, u2  ar , u3  ar , u4  ar , ...
2
un 
n
S n   ur 
r 1
3
1
1
7  8  10  11
2
2
AP or GP?
1 1 1 1
2  5  8  11
1  1.1  1.2  1.3
1  1.1  1.21  1.331
1 5 7 3
  
2 6 6 2
12  22  32  42
a  a 2  a3  a 4
1  1.1  1.11  1.111
1 1 1 1
  
2 6 12 36
3 9
  27  162
4 2
n  2n  3n  4n
2  4  8  16
1 1 1
1  
2 3 4
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 series
2
3
10
2
3
100
1 1 1
1
       ...    
4 4 4
4
1 1 1
1
       ...   
4 4 4
4
2
3
1 1 1
       ... 
4 4 4

2
3
4
1 1 1 1
1
          ... 
4 4 4  4
3
Infinite GPs: the formula
a  ar  ar 2  ar 3  ...  ar n 1 
n
a 1  r n 
1 r
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]