sequences and series

SEQUENCESSEQUENCESand and

SERIESSERIES

SEQUENCESSEQUENCES

Concept of sequences and series Concept of sequences and series is really study of patterns. is really study of patterns.

Patterns can be objects;Patterns can be objects;

Patterns can be objects;Patterns can be objects;

Nature;Nature;

Nature;Nature;

And numbers;And numbers;

And numbers; (Pascal triangle)And numbers; (Pascal triangle)

Sometimes it is easy to see Sometimes it is easy to see patterns and relationships in patterns and relationships in a string of numbers. For a string of numbers. For instance; instance;

2, 4, 6, 8, 10, 12, …2, 4, 6, 8, 10, 12, …

In the more difficult cases In the more difficult cases we need to use formula. we need to use formula. This topic teaches us how This topic teaches us how to use a logical approach in to use a logical approach in solving problems which solving problems which involves sequences and involves sequences and series.series.

Example; find 8Example; find 8thth term in the term in the given sequencegiven sequence

1, 4, 9, 16, 25, 36, ….1, 4, 9, 16, 25, 36, ….

SEQUENCES

Definition of Sequence: A pattern which is defined in the set of natural numbers is called a sequence.

Note: By the set of natural numbers we

mean all positive integers and denote this set by N.

That is, N = {1, 2, 3, ...}

We denote the first term by a1, the second term by a2, and so on.

Here, a1 is the first term

a2 is the second term

a3 is the third term ………………….......... an is the nth term or general term.

We can use another letter instead of letter a. For example, bn, cn, dn, etc. may also be the name for general term of a sequence.

A sequence is represented by (an) (an must be written inside brackets)

General term of a sequence is represented

by an (an must be written without brackets)

for the previous example, if we write the

general term, we use an = n2.

If we want to list the terms, we use (an) = (1, 4, 9, 16, ..., n2, ...)

Note:Note:An expression like a2.6 is nonsense since we cannot talk about 2.6th term. It is easy to realize that the definition for sequence prevents such potential mistakes. Clearly, expressions like a0, a–1 are also out of consideration.

Example: Example:

Write first five terms of the sequence whose general term is

1na n

Example: Example: Given the sequence with general term ,

find a5, a–2, a100

4 5

2n

na

n

Example: Example: Find the general term bn for the sequence whose first four terms are

1 2 3 4, , ,2 3 4 5

Example: Example: Write first five terms of the sequence whose general term is cn = (–1)n.

Example: Example: Find the general term an

for the sequence whose first four terms are 2, 4, 6, 8.

Example: Example: Given the sequence with general term bn = 2n + 3, find b5, b0, and b43.

Criteria for Existence of a Sequence

If there is at least one natural number which makes the general term undefined, then there is no such sequence.

Undefined: denominator is zero or even numbered root is less then zero.

Example: Example:

Is a general

term of a sequence? Why?

3 5

1n

na

n

Example: Example:

Is a general

term of a sequence? Why?

8na n

Example: Example:

Given xn = 2n + 5, which term of the sequence is equal to

A) 25 B) 17 C) 96

TYPES OF SEQUENCES Finite Sequence: If a sequence

contains countable number of terms, then it is a finite sequence.

Example; –10, –5, 0, 5, 10, 15, ..., 150

Infinite Sequence: If a sequence contains infinitely many terms, then it is an infinite sequence.

Example; 1, 1, 2, 3, 5, 8, ...

TYPES OF SEQUENCES Monotone Sequence: In general any

increasing or decreasing sequence is called monotone sequence.

If each term of a sequence is greater than the previous term, then that sequence is called an increasing sequence.

an+1 ≥ an

If each term of a sequence is less than the previous term, then that sequence is called a decreasing sequence.

an+1 < an

Example: Example: Prove that sequence (an) with general term an = 2n is an increasing sequence.

If an = 2n, then an+1 = 2(n + 1) = 2n + 2.

an+1 – an =

2n + 2 – 2n= 2. Since 2 > 0, (an) is an increasing

sequence.

Example: Example:

Prove that sequence (an) with

general term

is a decreasing sequence.

1

1na n

TYPES OF SEQUENCESPiecewise Sequences: If the general term of a sequence is defined by more than one formula, then it is called a piecewise sequence.

Example: Example: Write first four terms of the

sequence with general term

1,

2,

1

n

n is oddna

n is evenn

Example: Example:

Given the sequence with general term 2 5 , 10

8 , 10n

n n na

n n

a) find a20 b) find a1

c) which term is equal to 0?

TYPES OF SEQUENCES Recursively Defined Sequences: Sometimes terms in a sequence may depend on the other terms. Such a sequence is called a recursively defined sequence.

Example: Example:

Given a1 = 4 and an – 1 = an + 3

a) find a2

b) find the general term.

Example: Example:

Given f1 = 1, f2 = 2 ,

fn = fn – 2 + fn – 1 , find first six terms of the sequence.

ARITHMETIC SEQUENCES A sequence is arithmetic if the

differences between two consecutive terms are the same.

Let's look at the sequence 6, 10, 14, 18, …

Obviously the difference between each term is equal to 4

ARITHMETIC SEQUENCES

Definition: If a sequence (an) has the same difference d between its consecutive terms, then it is called as an arithmetic sequence.

ARITHMETIC SEQUENCES

(an) is arithmetic if an+1 = an + d such than n ∈ N, d ∈ R. Hence d is called as the common difference.

If d is positive, arithmetic sequence is increasing.

If d is negative, arithmetic sequence is decreasing.

Example: Example: State whether the following

sequences are arithmetic or not. If so, find the common difference.

7, 10, 13, 16, … 3, –2, –7, 12, … 1, 4, 9, 16, … 6, 6, 6, 6, …

Example: Example: State whether the following

sequences with general terms are arithmetic or not. If so, find the common difference.

an = 4n – 3 an = 2n an = n2 – n

ARITHMETIC SEQUENCES

General Term of an arithmetic sequence:

If an is arithmetic, then we only know that an+1 = an + d.

ARITHMETIC SEQUENCES

Let's write a few terms. a1

a2 = a1 + d

a3 = a2 + d = (a1 + d) + d = a1 + 2d

a4 = a3 + d = (a1 + 2d) + d = a1 + 3d

a5 = a1 + 4d .......... an = a1 + (n – 1)d

General term of an arithmetic sequence an with common difference d is

an = a1+(n – 1)d