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. .
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.
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.
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
Find the first order linear recurrence relation when:
U3 = 7, U4 = 15 and U5 = 31.
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.
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
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 !!!
c) Given the arithmetic sequence 2, 8, 14, 20, …. For what value of n is Un = 62?
2 a to e
3, 4, 6.
TJ Exercise 1 Questions 1 to 3
Find the sum of the first 15 terms of the arithmetic sequence which starts 3, 8, 13, 18, ……….
2, 10, 18, 26,…. First exceed 300?
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?
TJ Exercise 1 Questions 4 to 9
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:
3, 12, 48, …….
b) Find the geometric sequence whose 3rd term is 18 and whose 8th term is 4374
c) Given the geometric sequence 5, 10, 20, 40,…… find the value of n for which
TJ Exercise 2A Questions 1 to 4
Multiplying by r:
a) Find the sum to 6 terms of the geometric sequence whose first term is 6 and whose common ratio is 1.5.
What is the smallest value of n for which Sn>100?
Dividing we get:
TJ Exercise 2A Questions 5 to 7
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.
The sum to infinity for an arithmetic series is undefined.
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.
T.J. Exercise 2B
This is a geometric series with common ratio
a) Expand in ascending powers of x giving the first four terms.
c) Evaluate to 4 decimal places.
d) Expand in ascending powers of x giving the first four terms
Page 134 Exercise 7B Questions 2, 5.
TJ Exercise 3
(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
and so on to k = 10
and so on to k = 4
c) Express the following in notation. 1+4+7+10+…….+298
The sums of certain finite series can be found by a number of methods
We can use this to help evaluate many summation series.
TJ Exercise 4 and 5