Like this presentation? Why not share!

# Sequences and series

## by Mustafa Demirdag, Mathematics and Physics Trainer at FEZALAR on Jan 14, 2012

• 1,167 views

### Views

Total Views
1,167
Views on SlideShare
1,167
Embed Views
0

Likes
0
112
1

No embeds

### Categories

Uploaded via SlideShare as Microsoft PowerPoint

### Report content

11 of 1 previous next

• sumit1996 yaar file save nai hoti hai isme 1 year ago
Are you sure you want to

## Sequences and seriesPresentation Transcript

• SEQUENCES and SERIES
• SEQUENCES
• Concept of sequences and series is really study of patterns.
• Patterns can be objects;
• Patterns can be objects;
• Nature;
• Nature;
• And numbers;
• And numbers; (Pascal triangle)
• Sometimes it is easy to see patterns and relationships in a string of numbers. For instance;
• 2, 4, 6, 8, 10, 12, …
• In the more difficult cases we need to use formula. This topic teaches us how to use a logical approach in solving problems which involves sequences and series.
• Example; find 8 th term in the given sequence
• 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 a 1 , the second term by a 2 , and so on.
• Here, a 1 is the first term
• a 2 is the second term
• a 3 is the third term
• ………………… ..........
• a n is the n th term or general term.
• We can use another letter instead of letter a. For example, b n , c n , d n , etc. may also be the name for general term of a sequence.
• A sequence is represented by
• ( a n ) ( a n must be written inside brackets)
• General term of a sequence is represented by
• a n ( a n must be written without brackets)
• for the previous example, if we write the general term, we use a n = n 2 .
• If we want to list the terms, we use
• ( a n ) = (1, 4, 9, 16, ..., n 2 , ...)
• Note:
• An expression like a 2.6 is nonsense since we cannot talk about 2.6 th term. It is easy to realize that the definition for sequence prevents such potential mistakes. Clearly, expressions like a 0 , a –1 are also out of consideration.
• Example:
• Write first five terms of the sequence whose general term is
• Example:
• Given the sequence with general term ,
• find a 5 , a –2 , a 100
• Example:
• Find the general term b n for the sequence whose first four terms are
• Example:
• Write first five terms of the sequence whose general term is c n = (–1) n .
• Example:
• Find the general term a n for the sequence whose first four terms are 2, 4, 6, 8.
• Example:
• Given the sequence with general term b n = 2 n + 3, find b 5 , b 0 , and b 43 .
• 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:
• Is a general
• term of a sequence? Why?
• Example:
• Is a general
• term of a sequence? Why?
• Example:
• Given x n = 2 n + 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.
• a n +1 ≥ a n
• If each term of a sequence is less than the previous term, then that sequence is called a decreasing sequence.
• a n +1 < a n
• Example:
• Prove that sequence ( a n ) with general term a n = 2 n is an increasing sequence.
• If a n = 2 n , then a n +1 = 2( n + 1) = 2 n + 2.
• a n +1 – a n =
• 2 n + 2 – 2n= 2.
• Since 2 > 0, ( a n ) is an increasing sequence.
• Example:
• Prove that sequence ( a n ) with
• general term
• is a decreasing sequence.
• TYPES OF SEQUENCES
• Piecewise Sequences: If the general term of a sequence is defined by more than one formula, then it is called a piecewise sequence.
• Example:
• Write first four terms of the sequence with general term
• Example:
• Given the sequence with general term
a) find a 20 b) find a 1 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:
• Given a 1 = 4 and a n – 1 = a n + 3
• a) find a 2
• b) find the general term.
• Example:
• Given f 1 = 1, f 2 = 2 ,
• f n = f n – 2 + f n – 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 ( a n ) has the same difference d between its consecutive terms, then it is called as an arithmetic sequence.
• ARITHMETIC SEQUENCES
• ( a n ) is arithmetic if a n+ 1 = a n + 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:
• 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:
• State whether the following sequences with general terms are arithmetic or not. If so, find the common difference.
• a n = 4 n – 3
• a n = 2 n
• a n = n 2 – n
• ARITHMETIC SEQUENCES
• General Term of an arithmetic sequence:
• If a n is arithmetic, then we only know that a n +1 = a n + d .
• ARITHMETIC SEQUENCES
• Let's write a few terms.
• a 1
• a 2 = a 1 + d
• a 3 = a 2 + d = ( a 1 + d ) + d = a 1 + 2 d
• a 4 = a 3 + d = ( a 1 + 2 d ) + d = a 1 + 3 d
• a 5 = a 1 + 4 d
• ..........
• a n = a 1 + ( n – 1) d
• General term of an arithmetic sequence a n with common difference d is
• a n = a 1 +( n – 1) d