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

Sequences and Series