5.1 sequences

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5.1 sequences

  1. 1. Sequences
  2. 2. A sequence is an ordered list of infinitely many numbers that may or may not have a pattern. Sequences
  3. 3. A sequence is an ordered list of infinitely many numbers that may or may not have a pattern. Sequences Example A: 1, 3, 5, 7, 9,… is the sequence of odd numbers.
  4. 4. A sequence is an ordered list of infinitely many numbers that may or may not have a pattern. Sequences Example A: 1, 3, 5, 7, 9,… is the sequence of odd numbers. 1, 4, 9, 16, 25,… is the sequence of square numbers.
  5. 5. A sequence is an ordered list of infinitely many numbers that may or may not have a pattern. Sequences Example A: 1, 3, 5, 7, 9,… is the sequence of odd numbers. 1, 4, 9, 16, 25,… is the sequence of square numbers. 5, -2, , e2, -110, …is a sequence without an obvious pattern.
  6. 6. A sequence is an ordered list of infinitely many numbers that may or may not have a pattern. Definition: A sequence is the list of outputs f(n) of a function f where n = 1, 2, 3, 4, ….. We write f(n) as fn. Sequences Example A: 1, 3, 5, 7, 9,… is the sequence of odd numbers. 1, 4, 9, 16, 25,… is the sequence of square numbers. 5, -2, , e2, -110, …is a sequence without an obvious pattern.
  7. 7. A sequence is an ordered list of infinitely many numbers that may or may not have a pattern. Definition: A sequence is the list of outputs f(n) of a function f where n = 1, 2, 3, 4, ….. We write f(n) as fn. Sequences Example A: 1, 3, 5, 7, 9,… is the sequence of odd numbers. 1, 4, 9, 16, 25,… is the sequence of square numbers. 5, -2, , e2, -110, …is a sequence without an obvious pattern. A sequence may be listed as f1, f2 , f3 , …
  8. 8. A sequence is an ordered list of infinitely many numbers that may or may not have a pattern. Definition: A sequence is the list of outputs f(n) of a function f where n = 1, 2, 3, 4, ….. We write f(n) as fn. Sequences Example A: 1, 3, 5, 7, 9,… is the sequence of odd numbers. 1, 4, 9, 16, 25,… is the sequence of square numbers. 5, -2, , e2, -110, …is a sequence without an obvious pattern. A sequence may be listed as f1, f2 , f3 , … f100 = 100th number on the list,
  9. 9. A sequence is an ordered list of infinitely many numbers that may or may not have a pattern. Definition: A sequence is the list of outputs f(n) of a function f where n = 1, 2, 3, 4, ….. We write f(n) as fn. Sequences Example A: 1, 3, 5, 7, 9,… is the sequence of odd numbers. 1, 4, 9, 16, 25,… is the sequence of square numbers. 5, -2, , e2, -110, …is a sequence without an obvious pattern. A sequence may be listed as f1, f2 , f3 , … f100 = 100th number on the list, fn = the n’th number on the list,
  10. 10. A sequence is an ordered list of infinitely many numbers that may or may not have a pattern. Definition: A sequence is the list of outputs f(n) of a function f where n = 1, 2, 3, 4, ….. We write f(n) as fn. Sequences Example A: 1, 3, 5, 7, 9,… is the sequence of odd numbers. 1, 4, 9, 16, 25,… is the sequence of square numbers. 5, -2, , e2, -110, …is a sequence without an obvious pattern. A sequence may be listed as f1, f2 , f3 , … f100 = 100th number on the list, fn = the n’th number on the list, fn-1 = the (n – 1)’th number on the list or the number before fn.
  11. 11. Example B: a. For the sequence of square numbers 1, 4, 9, 16, … f3 = 9, f4= 16, f5 = 25, Sequences
  12. 12. Example B: a. For the sequence of square numbers 1, 4, 9, 16, … f3 = 9, f4= 16, f5 = 25, and a formula for fn is fn = n2. Sequences
  13. 13. Example B: a. For the sequence of square numbers 1, 4, 9, 16, … f3 = 9, f4= 16, f5 = 25, and a formula for fn is fn = n2. b. For the sequence of even numbers 2, 4, 6, 8, … f3= 6, f4 = 8, f5 = 10, Sequences
  14. 14. Example B: a. For the sequence of square numbers 1, 4, 9, 16, … f3 = 9, f4= 16, f5 = 25, and a formula for fn is fn = n2. b. For the sequence of even numbers 2, 4, 6, 8, … f3= 6, f4 = 8, f5 = 10, and a formula for fn is fn = 2*n Sequences
  15. 15. Example B: a. For the sequence of square numbers 1, 4, 9, 16, … f3 = 9, f4= 16, f5 = 25, and a formula for fn is fn = n2. b. For the sequence of even numbers 2, 4, 6, 8, … f3= 6, f4 = 8, f5 = 10, and a formula for fn is fn = 2*n Sequences c. For the sequence of odd numbers 1, 3, 5, … a general formula is fn= 2n – 1.
  16. 16. Example B: a. For the sequence of square numbers 1, 4, 9, 16, … f3 = 9, f4= 16, f5 = 25, and a formula for fn is fn = n2. b. For the sequence of even numbers 2, 4, 6, 8, … f3= 6, f4 = 8, f5 = 10, and a formula for fn is fn = 2*n Sequences c. For the sequence of odd numbers 1, 3, 5, … a general formula is fn= 2n – 1.
  17. 17. Example B: a. For the sequence of square numbers 1, 4, 9, 16, … f3 = 9, f4= 16, f5 = 25, and a formula for fn is fn = n2. b. For the sequence of even numbers 2, 4, 6, 8, … f3= 6, f4 = 8, f5 = 10, and a formula for fn is fn = 2*n Sequences c. For the sequence of odd numbers 1, 3, 5, … a general formula is fn= 2n – 1. d. For the sequence of odd numbers with alternating signs -1, 3, -5, 7, -9, …fn = (-1)n(2n – 1)
  18. 18. Example B: a. For the sequence of square numbers 1, 4, 9, 16, … f3 = 9, f4= 16, f5 = 25, and a formula for fn is fn = n2. b. For the sequence of even numbers 2, 4, 6, 8, … f3= 6, f4 = 8, f5 = 10, and a formula for fn is fn = 2*n Sequences c. For the sequence of odd numbers 1, 3, 5, … a general formula is fn= 2n – 1. d. For the sequence of odd numbers with alternating signs -1, 3, -5, 7, -9, …fn = (-1)n(2n – 1)
  19. 19. Example B: a. For the sequence of square numbers 1, 4, 9, 16, … f3 = 9, f4= 16, f5 = 25, and a formula for fn is fn = n2. b. For the sequence of even numbers 2, 4, 6, 8, … f3= 6, f4 = 8, f5 = 10, and a formula for fn is fn = 2*n Sequences c. For the sequence of odd numbers 1, 3, 5, … a general formula is fn= 2n – 1. d. For the sequence of odd numbers with alternating signs -1, 3, -5, 7, -9, …fn = (-1)n(2n – 1) A sequence whose signs alternate is called an alternating sequence as shown in part d.
  20. 20. Summation Notation
  21. 21. In mathematics, the Greek letter “ ” (sigma) means “to add”. Summation Notation
  22. 22. In mathematics, the Greek letter “ ” (sigma) means “to add”. Hence, “ x” means to add the x’s, Summation Notation
  23. 23. In mathematics, the Greek letter “ ” (sigma) means “to add”. Hence, “ x” means to add the x’s, “ (x*y)” means to add the x*y’s. Summation Notation
  24. 24. In mathematics, the Greek letter “ ” (sigma) means “to add”. Hence, “ x” means to add the x’s, “ (x*y)” means to add the x*y’s. (Of course the x's and xy's have to be given in the context.) Summation Notation
  25. 25. Summation Notation Given a list, let's say, of 100 numbers, f1, f2, f3,.., f100, In mathematics, the Greek letter “ ” (sigma) means “to add”. Hence, “ x” means to add the x’s, “ (x*y)” means to add the x*y’s. (Of course the x's and xy's have to be given in the context.)
  26. 26. Summation Notation Given a list, let's say, of 100 numbers, f1, f2, f3,.., f100, their sum f1 + f2 + f3 ... + f99 + f100 may be written in the - notation as: fkk = 1 100 In mathematics, the Greek letter “ ” (sigma) means “to add”. Hence, “ x” means to add the x’s, “ (x*y)” means to add the x*y’s. (Of course the x's and xy's have to be given in the context.)
  27. 27. Summation Notation Given a list, let's say, of 100 numbers, f1, f2, f3,.., f100, their sum f1 + f2 + f3 ... + f99 + f100 may be written in the - notation as: fkk = 1 100 A variable which is called the “index” variable, in this case k. In mathematics, the Greek letter “ ” (sigma) means “to add”. Hence, “ x” means to add the x’s, “ (x*y)” means to add the x*y’s. (Of course the x's and xy's have to be given in the context.)
  28. 28. Summation Notation Given a list, let's say, of 100 numbers, f1, f2, f3,.., f100, their sum f1 + f2 + f3 ... + f99 + f100 may be written in the - notation as: fkk = 1 100 A variable which is called the “index” variable, in this case k. k begins with the bottom number and counts up (runs) to the top number. In mathematics, the Greek letter “ ” (sigma) means “to add”. Hence, “ x” means to add the x’s, “ (x*y)” means to add the x*y’s. (Of course the x's and xy's have to be given in the context.)
  29. 29. Summation Notation Given a list, let's say, of 100 numbers, f1, f2, f3,.., f100, their sum f1 + f2 + f3 ... + f99 + f100 may be written in the - notation as: fkk = 1 100 A variable which is called the “index” variable, in this case k. k begins with the bottom number and counts up (runs) to the top number. The beginning number In mathematics, the Greek letter “ ” (sigma) means “to add”. Hence, “ x” means to add the x’s, “ (x*y)” means to add the x*y’s. (Of course the x's and xy's have to be given in the context.)
  30. 30. Summation Notation Given a list, let's say, of 100 numbers, f1, f2, f3,.., f100, their sum f1 + f2 + f3 ... + f99 + f100 may be written in the - notation as: fkk = 1 100 A variable which is called the “index” variable, in this case k. k begins with the bottom number and counts up (runs) to the top number. The beginning number The ending number In mathematics, the Greek letter “ ” (sigma) means “to add”. Hence, “ x” means to add the x’s, “ (x*y)” means to add the x*y’s. (Of course the x's and xy's have to be given in the context.)
  31. 31. Summation Notation Given a list, let's say, of 100 numbers, f1, f2, f3,.., f100, their sum f1 + f2 + f3 ... + f99 + f100 may be written in the - notation as: fk = f1 f2 f3 … f99 f100k = 1 100 A variable which is called the “index” variable, in this case k. k begins with the bottom number and counts up (runs) to the top number. The beginning number The ending number In mathematics, the Greek letter “ ” (sigma) means “to add”. Hence, “ x” means to add the x’s, “ (x*y)” means to add the x*y’s. (Of course the x's and xy's have to be given in the context.)
  32. 32. Summation Notation Given a list, let's say, of 100 numbers, f1, f2, f3,.., f100, their sum f1 + f2 + f3 ... + f99 + f100 may be written in the - notation as: fk = f1+ f2+ f3+ … + f99+ f100k = 1 100 A variable which is called the “index” variable, in this case k. k begins with the bottom number and counts up (runs) to the top number. The beginning number The ending number In mathematics, the Greek letter “ ” (sigma) means “to add”. Hence, “ x” means to add the x’s, “ (x*y)” means to add the x*y’s. (Of course the x's and xy's have to be given in the context.)
  33. 33. fk =k=4 8 ai = i=2 5 xjyj = j=6 9 aj = j=n n+3 Summation Notation
  34. 34. Example C: fk = f4+ f5+ f6+ f7+ f8k=4 8 ai = i=2 5 xjyj = j=6 9 aj = j=n n+3 Summation Notation
  35. 35. Example C: fk = f4+ f5+ f6+ f7+ f8k=4 8 ai = a2+ a3+ a4+ a5i=2 5 xjyj = j=6 9 aj = j=n n+3 Summation Notation
  36. 36. Example C: fk = f4+ f5+ f6+ f7+ f8k=4 8 ai = a2+ a3+ a4+ a5i=2 5 xjyj = x6y6+ x7y7+ x8y8+ x9y9j=6 9 aj = j=n n+3 Summation Notation
  37. 37. Example C: fk = f4+ f5+ f6+ f7+ f8k=4 8 ai = a2+ a3+ a4+ a5i=2 5 xjyj = x6y6+ x7y7+ x8y8+ x9y9j=6 9 aj = an+ an+1+ an+2+ an+3j=n n+3 Summation Notation
  38. 38. Example C: fk = f4+ f5+ f6+ f7+ f8k=4 8 ai = a2+ a3+ a4+ a5i=2 5 xjyj = x6y6+ x7y7+ x8y8+ x9y9j=6 9 aj = an+ an+1+ an+2+ an+3j=n n+3 Summation Notation Summation notation are used to express formulas in mathematics.
  39. 39. Example C: fk = f4+ f5+ f6+ f7+ f8k=4 8 ai = a2+ a3+ a4+ a5i=2 5 xjyj = x6y6+ x7y7+ x8y8+ x9y9j=6 9 aj = an+ an+1+ an+2+ an+3j=n n+3 Summation Notation Summation notation are used to express formulas in mathematics. An example is the formula for averaging.
  40. 40. Example C: fk = f4+ f5+ f6+ f7+ f8k=4 8 ai = a2+ a3+ a4+ a5i=2 5 xjyj = x6y6+ x7y7+ x8y8+ x9y9j=6 9 aj = an+ an+1+ an+2+ an+3j=n n+3 Summation Notation Summation notation are used to express formulas in mathematics. An example is the formula for averaging. Given n numbers, f1, f2, f3,.., fn, their average (mean), written as f, is (f1 + f2 + f3 ... + fn-1 + fn)/n.
  41. 41. Example C: fk = f4+ f5+ f6+ f7+ f8k=4 8 ai = a2+ a3+ a4+ a5i=2 5 xjyj = x6y6+ x7y7+ x8y8+ x9y9j=6 9 aj = an+ an+1+ an+2+ an+3j=n n+3 Summation Notation Summation notation are used to express formulas in mathematics. An example is the formula for averaging. Given n numbers, f1, f2, f3,.., fn, their average (mean), written as f, is (f1 + f2 + f3 ... + fn-1 + fn)/n. In notation, f = k=1 n fk n
  42. 42. The index variable is also used as the variable that generates the numbers to be summed. Summation Notation
  43. 43. Example D: a. (k2 – 1) k=5 8 The index variable is also used as the variable that generates the numbers to be summed. Summation Notation
  44. 44. Example D: a. (k2 – 1) = k=5 8 The index variable is also used as the variable that generates the numbers to be summed. k=5 k=6 k=7 k=8 Summation Notation
  45. 45. Example D: a. (k2 – 1) = (52–1) + (62–1) + (72–1) + (82–1) k=5 8 The index variable is also used as the variable that generates the numbers to be summed. k=5 k=6 k=7 k=8 Summation Notation
  46. 46. Example D: a. (k2 – 1) = (52–1) + (62–1) + (72–1) + (82–1) = 24 + 35 + 48 + 63 k=5 8 The index variable is also used as the variable that generates the numbers to be summed. k=5 k=6 k=7 k=8 Summation Notation
  47. 47. a. (k2 – 1) = (52–1) + (62–1) + (72–1) + (82–1) = 24 + 35 + 48 + 63 = 170 k=5 8 The index variable is also used as the variable that generates the numbers to be summed. k=5 k=6 k=7 k=8 Summation Notation Example D:
  48. 48. a. (k2 – 1) = (52–1) + (62–1) + (72–1) + (82–1) = 24 + 35 + 48 + 63 = 170 k=5 8 The index variable is also used as the variable that generates the numbers to be summed. k=5 k=6 k=7 k=8 Summation Notation b. (-1)k(3k + 2) k=3 5 Example D:
  49. 49. a. (k2 – 1) = (52–1) + (62–1) + (72–1) + (82–1) = 24 + 35 + 48 + 63 = 170 k=5 8 The index variable is also used as the variable that generates the numbers to be summed. k=5 k=6 k=7 k=8 Summation Notation b. (-1)k(3k + 2) =(-1)3(3*3+2)+(-1)4(3*4+2)+(-1)5(3*5+2)k=3 5 Example D:
  50. 50. a. (k2 – 1) = (52–1) + (62–1) + (72–1) + (82–1) = 24 + 35 + 48 + 63 = 170 k=5 8 The index variable is also used as the variable that generates the numbers to be summed. k=5 k=6 k=7 k=8 Summation Notation b. (-1)k(3k + 2) =(-1)3(3*3+2)+(-1)4(3*4+2)+(-1)5(3*5+2) = -11 + 14 – 17 k=3 5 Example D:
  51. 51. a. (k2 – 1) = (52–1) + (62–1) + (72–1) + (82–1) = 24 + 35 + 48 + 63 = 170 k=5 8 The index variable is also used as the variable that generates the numbers to be summed. k=5 k=6 k=7 k=8 Summation Notation b. (-1)k(3k + 2) =(-1)3(3*3+2)+(-1)4(3*4+2)+(-1)5(3*5+2) = -11 + 14 – 17 = -14 k=3 5 Example D:
  52. 52. a. (k2 – 1) = (52–1) + (62–1) + (72–1) + (82–1) = 24 + 35 + 48 + 63 = 170 k=5 8 The index variable is also used as the variable that generates the numbers to be summed. k=5 k=6 k=7 k=8 Summation Notation b. (-1)k(3k + 2) =(-1)3(3*3+2)+(-1)4(3*4+2)+(-1)5(3*5+2) = -11 + 14 – 17 = -14 k=3 5 In part b, the multiple (-1)k change the sums to an alternating sum, t Example D:
  53. 53. a. (k2 – 1) = (52–1) + (62–1) + (72–1) + (82–1) = 24 + 35 + 48 + 63 = 170 k=5 8 The index variable is also used as the variable that generates the numbers to be summed. k=5 k=6 k=7 k=8 Summation Notation b. (-1)k(3k + 2) =(-1)3(3*3+2)+(-1)4(3*4+2)+(-1)5(3*5+2) = -11 + 14 – 17 = -14 k=3 5 In part b, the multiple (-1)k change the sums to an alternating sum, that is, a sum where the terms alternate between positive and negative numbers. Example D:
  54. 54. Exercise A. List the first four terms of each of the following sequences given by fn where n = 1,2, 3, .. Sequences 2. 4. 5. 6. 7. 8. 9. 10. 1. 3. fn = –5 + n fn = 5 – n fn = 3n fn = –5 + 2n fn = 5 – n2 fn = –4n + 1 fn = –5 + nfn = (–1)n5 / n fn = (3n + 2)/(–1 – n) fn = 2n2 – n fn = n2 / (2n + 1)
  55. 55. Exercise B. Find a formula fn for each of the following sequences. Sequences 12. 14. 15. 16. 17. 18. 19. 20. 11. 13. 2, 3, 4, 5.. –3, –2, –1, 0, 1.. 10, 20, 30, 40,.. 5, 10, 15, 20,.. –40, –30, –20, –10, 0,.. –5, –10, –15, –20,.. 1/2, 1/3, 1/4, 1/5.. 1/2, –2/3, 3/4, –4/5.. –1, 1/4, –1/9, 1/16, –1/25,.. 1, r1, r2, r3,..
  56. 56. Sequences

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