Sequences and series

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Sequences and series

  1. 1. SEQUENCES and SERIES
  2. 2. SEQUENCES <ul><li>Concept of sequences and series is really study of patterns. </li></ul>
  3. 3. <ul><li>Patterns can be objects; </li></ul>
  4. 4. <ul><li>Patterns can be objects; </li></ul>
  5. 5. <ul><li>Nature; </li></ul>
  6. 6. <ul><li>Nature; </li></ul>
  7. 7. <ul><li>And numbers; </li></ul>
  8. 8. <ul><li>And numbers; (Pascal triangle) </li></ul>
  9. 9. <ul><li>Sometimes it is easy to see patterns and relationships in a string of numbers. For instance; </li></ul><ul><li>2, 4, 6, 8, 10, 12, … </li></ul>
  10. 10. <ul><li>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. </li></ul>
  11. 11. <ul><li>Example; find 8 th term in the given sequence </li></ul><ul><li>1, 4, 9, 16, 25, 36, …. </li></ul>
  12. 12. SEQUENCES <ul><li>Definition of Sequence: A pattern which is defined in the set of natural numbers is called a sequence. </li></ul><ul><li>Note: </li></ul><ul><li>By the set of natural numbers we mean all positive integers and denote this set by N. </li></ul><ul><li>That is, N = {1, 2, 3, ...} </li></ul>
  13. 13. <ul><li>We denote the first term by a 1 , the second term by a 2 , and so on. </li></ul><ul><li>Here, a 1 is the first term </li></ul><ul><li>a 2 is the second term </li></ul><ul><li>a 3 is the third term </li></ul><ul><li>………………… .......... </li></ul><ul><li>a n is the n th term or general term. </li></ul><ul><li>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. </li></ul>
  14. 14. <ul><li>A sequence is represented by </li></ul><ul><li>( a n ) ( a n must be written inside brackets) </li></ul><ul><li>General term of a sequence is represented by </li></ul><ul><li>a n ( a n must be written without brackets) </li></ul><ul><li>for the previous example, if we write the general term, we use a n = n 2 . </li></ul><ul><li>If we want to list the terms, we use </li></ul><ul><li>( a n ) = (1, 4, 9, 16, ..., n 2 , ...) </li></ul>
  15. 15. Note: <ul><li>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. </li></ul>
  16. 16. Example: <ul><li>Write first five terms of the sequence whose general term is </li></ul>
  17. 17. Example: <ul><li>Given the sequence with general term , </li></ul><ul><li>find a 5 , a –2 , a 100 </li></ul>
  18. 18. Example: <ul><li>Find the general term b n for the sequence whose first four terms are </li></ul>
  19. 19. Example: <ul><li>Write first five terms of the sequence whose general term is c n = (–1) n . </li></ul>
  20. 20. Example: <ul><li>Find the general term a n for the sequence whose first four terms are 2, 4, 6, 8. </li></ul>
  21. 21. Example: <ul><li>Given the sequence with general term b n = 2 n + 3, find b 5 , b 0 , and b 43 . </li></ul>
  22. 22. Criteria for Existence of a Sequence <ul><li>If there is at least one natural number which makes the general term undefined , then there is no such sequence. </li></ul><ul><li>Undefined: denominator is zero or even numbered root is less then zero. </li></ul>
  23. 23. Example: <ul><li>Is a general </li></ul><ul><li>term of a sequence? Why? </li></ul>
  24. 24. Example: <ul><li>Is a general </li></ul><ul><li>term of a sequence? Why? </li></ul>
  25. 25. Example: <ul><li>Given x n = 2 n + 5, which term of the sequence is equal to </li></ul><ul><li>A) 25 B) 17 C) 96 </li></ul>
  26. 26. TYPES OF SEQUENCES <ul><li>Finite Sequence: If a sequence contains countable number of terms, then it is a finite sequence. </li></ul><ul><li>Example; –10, –5, 0, 5, 10, 15, ..., 150 </li></ul><ul><li>Infinite Sequence: If a sequence contains infinitely many terms, then it is an infinite sequence. </li></ul><ul><li>Example; 1, 1, 2, 3, 5, 8, ... </li></ul>
  27. 27. TYPES OF SEQUENCES <ul><li>Monotone Sequence: In general any increasing or decreasing sequence is called monotone sequence. </li></ul><ul><li>If each term of a sequence is greater than the previous term, then that sequence is called an increasing sequence. </li></ul><ul><li>a n +1 ≥ a n </li></ul><ul><li>If each term of a sequence is less than the previous term, then that sequence is called a decreasing sequence. </li></ul><ul><li>a n +1 < a n </li></ul>
  28. 28. Example: <ul><li>Prove that sequence ( a n ) with general term a n = 2 n is an increasing sequence. </li></ul><ul><li>If a n = 2 n , then a n +1 = 2( n + 1) = 2 n + 2. </li></ul><ul><li>a n +1 – a n = </li></ul><ul><li>2 n + 2 – 2n= 2. </li></ul><ul><li>Since 2 > 0, ( a n ) is an increasing sequence. </li></ul>
  29. 29. Example: <ul><li>Prove that sequence ( a n ) with </li></ul><ul><li>general term </li></ul><ul><li>is a decreasing sequence. </li></ul>
  30. 30. TYPES OF SEQUENCES <ul><li>Piecewise Sequences: If the general term of a sequence is defined by more than one formula, then it is called a piecewise sequence. </li></ul>
  31. 31. Example: <ul><li>Write first four terms of the sequence with general term </li></ul>
  32. 32. Example: <ul><li>Given the sequence with general term </li></ul>a) find a 20 b) find a 1 c) which term is equal to 0?
  33. 33. TYPES OF SEQUENCES <ul><li>Recursively Defined Sequences: Sometimes terms in a sequence may depend on the other terms. Such a sequence is called a recursively defined sequence. </li></ul>
  34. 34. Example: <ul><li>Given a 1 = 4 and a n – 1 = a n + 3 </li></ul><ul><li>a) find a 2 </li></ul><ul><li>b) find the general term. </li></ul>
  35. 35. Example: <ul><li>Given f 1 = 1, f 2 = 2 , </li></ul><ul><li>f n = f n – 2 + f n – 1 , find first six terms of the sequence. </li></ul>
  36. 36. ARITHMETIC SEQUENCES <ul><li>A sequence is arithmetic if the differences between two consecutive terms are the same. </li></ul><ul><li>Let's look at the sequence </li></ul><ul><li>6, 10, 14, 18, … </li></ul><ul><li>Obviously the difference between each term is equal to 4 </li></ul>
  37. 37. ARITHMETIC SEQUENCES <ul><li>Definition: If a sequence ( a n ) has the same difference d between its consecutive terms, then it is called as an arithmetic sequence. </li></ul>
  38. 38. ARITHMETIC SEQUENCES <ul><li>( 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. </li></ul><ul><li>If d is positive, arithmetic sequence is increasing. </li></ul><ul><li>If d is negative, arithmetic sequence is decreasing. </li></ul>
  39. 39. Example: <ul><li>State whether the following sequences are arithmetic or not. If so, find the common difference. </li></ul><ul><li>7, 10, 13, 16, … </li></ul><ul><li>3, –2, –7, 12, … </li></ul><ul><li>1, 4, 9, 16, … </li></ul><ul><li>6, 6, 6, 6, … </li></ul>
  40. 40. Example: <ul><li>State whether the following sequences with general terms are arithmetic or not. If so, find the common difference. </li></ul><ul><li>a n = 4 n – 3 </li></ul><ul><li>a n = 2 n </li></ul><ul><li>a n = n 2 – n </li></ul>
  41. 41. ARITHMETIC SEQUENCES <ul><li>General Term of an arithmetic sequence: </li></ul><ul><li>If a n is arithmetic, then we only know that a n +1 = a n + d . </li></ul>
  42. 42. ARITHMETIC SEQUENCES <ul><li>Let's write a few terms. </li></ul><ul><li>a 1 </li></ul><ul><li>a 2 = a 1 + d </li></ul><ul><li>a 3 = a 2 + d = ( a 1 + d ) + d = a 1 + 2 d </li></ul><ul><li>a 4 = a 3 + d = ( a 1 + 2 d ) + d = a 1 + 3 d </li></ul><ul><li>a 5 = a 1 + 4 d </li></ul><ul><li>.......... </li></ul><ul><li>a n = a 1 + ( n – 1) d </li></ul>
  43. 43. <ul><li>General term of an arithmetic sequence a n with common difference d is </li></ul><ul><li>a n = a 1 +( n – 1) d </li></ul>

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