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Sequences and Series. A sequence is an ordered list of numbers where each term is obtained according to a fixed rule. . A series, or progression, is a sum. The terms of which form a sequence. . The n th term of a sequence is often denoted U n , so that, for example, U is the first term. .

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sequences and series
Sequences and Series

A sequence is an ordered list of numbers where each term is obtained according to a fixed rule.

A series, or progression, is a sum. The terms of which form a sequence.

The nth term of a sequence is often denoted Un, so that, for example, U is the first term.

A sequence can be defined by a recurrence relation where Un+1 is given as a function of lower, earlier terms.

slide2

A first – order recurrence relation is where Un+1=rUn + d, where r and d are constants. This relation is linear.

A sequence can be defined by a formula for Un, given as a function.

Un = f(n)

Being given the first few terms of a sequence is not enough to identify the sequence.

slide3

For example. Identify the next term in the sequence

1, 2, 3, …, …,

Possible answers include:

If however we also know that the sequence is generated by a first order linear recurrence relation, then we know

slide5

Example 1

Find the first order linear recurrence relation when:

U3 = 7, U4 = 15 and U5 = 31.

slide6

Note

Given a relation

When this repetition happens, Un is referred to as a fixed point.

In this case, for any other value of Un, the relation generates values that move away or diverge from the value of 2.

Un =2 is an unstable fixed point.

slide7

Given the relation , then if for some value of n,

Un = 4, the sequence would proceed 4, 4, 4, 4, ……

If any other value of Un is used apart from 4, the relation generates terms whose value moves towards or converges on 4.

Un = 4 is a stable fixed point, often referred to as the limit of the recurrence relation.

In general, for the relation , we have a fixed point when

arithmetic sequences
Arithmetic Sequences

If a sequence is generated so that, for all n,

then the sequence is known as an arithmetic sequence. The constant d is referred to as the common difference.

This is a first order linear recurrence relation.

Traditionally, U1 is represented by the letter a: U1 = a.

This can be proved by induction – LATER !!!

slide9

Find the nth term

  • The 10th term of the arithmetic sequence 6, 11, 16, ……..
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b) Find the arithmetic sequence for which U3 = 9 and U7 = 17.

Subtracting gives:

Substituting gives:

slide12

Page 117 Exercise 2A Questions 1 a, c, d

2 a to e

3, 4, 6.

TJ Exercise 1 Questions 1 to 3

slide14

Find the sum of the first 15 terms of the arithmetic sequence which starts 3, 8, 13, 18, ……….

slide15

When does the sum of the arithmetic sequence which starts

2, 10, 18, 26,…. First exceed 300?

We require

Solving

we get

slide16

The sum of the first four terms of an arithmetic sequence is 26. The sum of the first twelve terms is 222. What is the sum of the first 20 terms?

NOTE

geometric sequences
Geometric Sequences

If a sequence is generated so that for all

then the sequence is known as a geometric sequence. The constant r is referred to as the common ratio.

The nth term:

slide19

a) Find the nth term and the 10th term of the geometric sequence

3, 12, 48, …….

b) Find the geometric sequence whose 3rd term is 18 and whose 8th term is 4374

Substituting gives

the sum to n terms of a geometric sequence
The Sum to n Terms of a Geometric Sequence

PROOF

Multiplying by r:

Subtracting:

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a) Find the sum to 6 terms of the geometric sequence whose first term is 6 and whose common ratio is 1.5.

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b) A geometric sequence starts 12, 15, 18.75,……

What is the smallest value of n for which Sn>100?

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A geometric series is such that S3 = 14 and S6 = 126.

  • Identify the series.

Dividing we get:

slide27

Page 127 Exercise 5a Questions 1, 2, 3, 4,

TJ Exercise 2A Questions 5 to 7

infinite series partial sums sum to infinity
Infinite Series, Partial Sums, Sum to infinity.

An Infinite series is a series which has an infinite number of terms.

When we have an infinite series then Sn is defined as the sum to n terms of that series. Such a sum is referred to as a partial sum of the series.

If the partial sum, Sn, tends towards a limit as n tends to infinity, then the limit is called the sum to infinity of the series.

slide29

Arithmetic Series

The sum to infinity for an arithmetic series is undefined.

slide31

Find the sum to infinity of the geometric series

  • 24 + 12 + 6 + …….. If it exists.
slide33

c) Given that 12 and 3 are two adjacent terms of an infinite geometric progression with find the first term.

Hence the first term is 48.

slide37

Now Consider

This is a geometric series with common ratio

slide39

b) Expand giving the first four terms.

c) Evaluate to 4 decimal places.

slide41

Page 134 Exercise 7A Questions 2, 4.

Page 134 Exercise 7B Questions 2, 5.

TJ Exercise 3

summation of a series

The Sigma Notation

(i.e. the sum of all k2 for k = 1 to k = n)

In general is the series with the first term f(1), second term

f(2), third term f(3) and last term f(n)

Summation of a Series

The sigma notation is used as a more concise way of writing a series.

e.g. 12 + 22 + 32 + 42 + 52 +…………+n2 can be written more concisely as

summation of a series48
Summation of a Series

The sums of certain finite series can be found by a number of methods

Proof:

We can use this to help evaluate many summation series.