30: Sequences and 30: Sequences and SeriesSeries
© Christine Crisp
““Teach A Level Maths”Teach A Level Maths”
Vol. 1: AS Core Vol. 1: AS Core ModulesModules
Sequences and Series
Module C1
AQAEdexcel
OCR
MEI/OCR
Module C2
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Sequences and Series
Examples of Sequences
e.g. 1 ...,8,6,4,2
e.g. 2 ...,4
1,
3
1,
2
1,1
e.g. 3 ...,64,16,4,1
A sequence is an ordered list of numbers
The 3 dots are used to show that a sequence continues
Sequences and SeriesRecurrence
Relations
...,9,7,5,3
Can you predict the next term of the sequence
?Suppose the formula continues by adding 2 to each term.The formula that generates the sequence is then
21 nn uu
223 uu
where and are terms of the sequencenu 1nu
is the 1st term, so1u 31 u5232 u
7253 u
etc.
1n 212 uu
2n
11
Sequences and SeriesRecurrence
Relations
nn uu 41
e.g. 1 Give the 1st term and write down a
recurrence relation for the sequence...,64,16,4,1
1st term: 11 uSolution:
Other letters may be used instead of u and n, so the formula could, for example, be given as
kk aa 41
Recurrence relation:
A formula such as is called a
recurrence relation
21 nn uu
Sequences and SeriesRecurrence
Relationse.g. 2 Write down the 2nd, 3rd and 4th terms of
the sequence given by 32,5 11 ii uuu
1iSolution: 32 12 uu
73)5(22 u
2i 32 23 uu
113)7(23 u
3i 32 34 uu
193)11(24 uThe sequence
is ...,19,11,7,5
Sequences and SeriesProperties of
sequencesConvergent sequences approach a
certain value
e.g. approaches 2...1,1,1,1,11615
87
43
21
n
nu
Sequences and SeriesProperties of
sequences
e.g. approaches 0...,,,,,1161
81
41
21
This convergent sequence also
oscillates
Convergent sequences approach a
certain value
n
nu
Sequences and SeriesProperties of
sequences
e.g. ...,10,8,6,4,2
Divergent sequences do not
converge
n
nu
Sequences and SeriesProperties of
sequences
e.g. ...,16,8,4,2,1
This divergent sequence also
oscillates
Divergent sequences do not
converge
n
nu
Sequences and SeriesProperties of
sequences
e.g
.
...,3,2,1,3,2,1,3,2,1
This divergent sequence is also
periodic
Divergent sequences do not
converge
n
nu
Sequences and SeriesConvergent
ValuesIt is not always easy to see what value a
sequence converges to. e.g.
n
nn u
uuu
310,1 11
...,11
103,
7
11,7,1
The sequence
isTo find the value that the sequence converges to we use the fact that eventually ( at infinity! ) the ( n + 1 ) th term equals the n th term.
Let . Then, uuu nn 1 u
uu
310
01032 uu
0)2)(5( uu 25 uu since
uu 3102 Multiply by u :
Sequences and Series
Exercises1. Write out the first 5 terms of the following sequences and describe the sequence using the words convergent, divergent, oscillating, periodic as appropriate
(b) n
n uuu
12 11 and
2. What value does the sequence given by
,u 21
34 11 nn uuu and (a)
nn u uu 21
11 16 and (c)
Ans: 8,5,2,1,4 Divergent
Ans:
2,,2,,221
21 Divergent
Periodic
Ans: 1,2,4,8,16 Convergent Oscillating
uuu nn 1Let
370330 uuu7
30 u
to? converge 3301 nn uu
Sequences and SeriesGeneral Term of a
SequenceSome sequences can also be defined by giving a general term. This general term is usually called the nth term.
n2
n
1
The general term can easily be checked by substituting n = 1, n = 2, etc.
e.g. 1
nu...,8,6,4,2
e.g. 2 nu...,4
1,
3
1,
2
1,1
e.g. 3 nu...,64,16,4,1 1)4( n
Sequences and SeriesExercise
sWrite out the first 5 terms of the following sequences
1.
(b)
nnu )2(
nun 41 (a)
22nun (c) n
nu )1((d)
19,15,11,7,3 32,16,8,4,2
50,32,18,8,2
1,1,1,1,1 Give the general term of each of the following sequences
2.
...,7,5,3,1(a
) 12 nun
...,243,81,27,9,3 (c)
(b)
...,25,16,9,4,1
(d)
...,5,5,5,5,5 5)1( 1 nnu
2nun n
nu )3(
Sequences and SeriesSeries
When the terms of a sequence are added, we get a series
...,25,16,9,4,1The sequencegives the series
...2516941
Sigma Notation for a SeriesA series can be described using the general
term100...2516941 e.g.
10
1
2ncan be written
is the Greek capital letter S, used for Sum
1st value of n
last value of n
Sequences and Series
16...8642 (a)
8
1
2n
1003...2793 (b)
2. Write the following using sigma notation
Exercises1. Write out the first 3 terms and the last term of the series given below in sigma notation
20
1
12n(a
) 1
1024...842 (b)
10
1
2 n
3n = 1n = 2
39...5
100
1
3 n
n = 20