The document discusses sequences and series. It defines what a sequence is, including the general term and different types of sequences such as arithmetic, finite, infinite, monotone, and piecewise sequences. It also defines arithmetic sequences specifically and provides the general term for an arithmetic sequence as an = a1 + (n - 1)d, where d is the common difference. Examples are given throughout to illustrate sequence concepts and properties.

1. SEQUENCES and SERIES

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

3. Patterns can be objects;

4. Patterns can be objects;

5. Nature;

6. Nature;

7. And numbers;

8. And numbers; (Pascal triangle)

9. Sometimes it is easy to see patterns and relationships in a string of numbers. For instance; 2, 4, 6, 8, 10, 12, …

10. 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.

11. Example; find 8 th term in the given sequence 1, 4, 9, 16, 25, 36, ….

12. 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, ...}

13. 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.

14. 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 , ...)

15. 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.

16. Example: Write first five terms of the sequence whose general term is

17. Example: Given the sequence with general term , find a 5 , a –2 , a 100

18. Example: Find the general term b n for the sequence whose first four terms are

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

20. Example: Find the general term a n for the sequence whose first four terms are 2, 4, 6, 8.

21. Example: Given the sequence with general term b n = 2 n + 3, find b 5 , b 0 , and b 43 .

22. 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.

23. Example: Is a general term of a sequence? Why?

24. Example: Is a general term of a sequence? Why?

25. Example: Given x n = 2 n + 5, which term of the sequence is equal to A) 25 B) 17 C) 96

26. 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, ...

27. 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

28. 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.

29. Example: Prove that sequence ( a n ) with general term is a decreasing sequence.

30. 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.

31. Example: Write first four terms of the sequence with general term

32. Example: Given the sequence with general term a) find a 20 b) find a 1 c) which term is equal to 0?

33. 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.

34. Example: Given a 1 = 4 and a n – 1 = a n + 3 a) find a 2 b) find the general term.

35. Example: Given f 1 = 1, f 2 = 2 , f n = f n – 2 + f n – 1 , find first six terms of the sequence.

36. 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

37. 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.

38. 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.

39. 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, …

40. 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

41. ARITHMETIC SEQUENCES General Term of an arithmetic sequence: If a n is arithmetic, then we only know that a n +1 = a n + d .

42. 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

43. General term of an arithmetic sequence a n with common difference d is a n = a 1 +( n – 1) d