5.1 sequences and summation notation

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5.1 sequences and summation notation

  1. 1. Sequences<br />
  2. 2. Sequences<br />A sequence is an ordered list of infinitely many numbers that may or may not have a pattern.<br />
  3. 3. Sequences<br />A sequence is an ordered list of infinitely many numbers that may or may not have a pattern.<br />Example A:<br />1, 3, 5, 7, 9,… is the sequence of odd numbers.<br />
  4. 4. Sequences<br />A sequence is an ordered list of infinitely many numbers that may or may not have a pattern.<br />Example A:<br />1, 3, 5, 7, 9,… is the sequence of odd numbers.<br />1, 4, 9, 16, 25,… is the sequence of square numbers.<br />
  5. 5. Sequences<br />A sequence is an ordered list of infinitely many numbers that may or may not have a pattern.<br />Example A:<br />1, 3, 5, 7, 9,… is the sequence of odd numbers.<br />1, 4, 9, 16, 25,… is the sequence of square numbers.<br />5, -2, , e2, -110, …is a sequence without an obvious pattern. <br />
  6. 6. Sequences<br />A sequence is an ordered list of infinitely many numbers that may or may not have a pattern.<br />Example A:<br />1, 3, 5, 7, 9,… is the sequence of odd numbers.<br />1, 4, 9, 16, 25,… is the sequence of square numbers.<br />5, -2, , e2, -110, …is a sequence without an obvious pattern. <br />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.<br />
  7. 7. Sequences<br />A sequence is an ordered list of infinitely many numbers that may or may not have a pattern.<br />Example A:<br />1, 3, 5, 7, 9,… is the sequence of odd numbers.<br />1, 4, 9, 16, 25,… is the sequence of square numbers.<br />5, -2, , e2, -110, …is a sequence without an obvious pattern. <br />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.<br />A sequence may be listed as f1, f2 , f3 , … <br />
  8. 8. Sequences<br />A sequence is an ordered list of infinitely many numbers that may or may not have a pattern.<br />Example A:<br />1, 3, 5, 7, 9,… is the sequence of odd numbers.<br />1, 4, 9, 16, 25,… is the sequence of square numbers.<br />5, -2, , e2, -110, …is a sequence without an obvious pattern. <br />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.<br />A sequence may be listed as f1, f2 , f3 , … <br />f100 = 100th number on the list, <br />
  9. 9. Sequences<br />A sequence is an ordered list of infinitely many numbers that may or may not have a pattern.<br />Example A:<br />1, 3, 5, 7, 9,… is the sequence of odd numbers.<br />1, 4, 9, 16, 25,… is the sequence of square numbers.<br />5, -2, , e2, -110, …is a sequence without an obvious pattern. <br />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.<br />A sequence may be listed as f1, f2 , f3 , … <br />f100 = 100th number on the list, <br />fn = the n’th number on the list,<br />
  10. 10. Sequences<br />A sequence is an ordered list of infinitely many numbers that may or may not have a pattern.<br />Example A:<br />1, 3, 5, 7, 9,… is the sequence of odd numbers.<br />1, 4, 9, 16, 25,… is the sequence of square numbers.<br />5, -2, , e2, -110, …is a sequence without an obvious pattern. <br />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.<br />A sequence may be listed as f1, f2 , f3 , … <br />f100 = 100th number on the list, <br />fn = the n’th number on the list,<br />fn-1 = the (n – 1)’th number on the list or the number before fn. <br />
  11. 11. Sequences<br />Example B: <br />a. For the sequence of square numbers 1, 4, 9, 16, … <br /> f3 = 9, f4= 16, f5 = 25, <br />
  12. 12. Sequences<br />Example B: <br />a. For the sequence of square numbers 1, 4, 9, 16, … <br /> f3 = 9, f4= 16, f5 = 25, <br /> and a formula for fn is fn = n2.<br />
  13. 13. Sequences<br />Example B: <br />a. For the sequence of square numbers 1, 4, 9, 16, … <br /> f3 = 9, f4= 16, f5 = 25, <br /> and a formula for fn is fn = n2.<br />b. For the sequence of even numbers 2, 4, 6, 8, … <br /> f3= 6, f4 = 8, f5 = 10, <br />
  14. 14. Sequences<br />Example B: <br />a. For the sequence of square numbers 1, 4, 9, 16, … <br /> f3 = 9, f4= 16, f5 = 25, <br /> and a formula for fn is fn = n2.<br />b. For the sequence of even numbers 2, 4, 6, 8, … <br /> f3= 6, f4 = 8, f5 = 10, <br /> and a formula for fn is fn = 2*n <br />
  15. 15. Sequences<br />Example B: <br />a. For the sequence of square numbers 1, 4, 9, 16, … <br /> f3 = 9, f4= 16, f5 = 25, <br /> and a formula for fn is fn = n2.<br />b. For the sequence of even numbers 2, 4, 6, 8, … <br /> f3= 6, f4 = 8, f5 = 10, <br /> and a formula for fn is fn = 2*n <br />c. For the sequence of odd numbers 1, 3, 5, …<br /> a general formula is fn= 2n – 1.<br />
  16. 16. Sequences<br />Example B: <br />a. For the sequence of square numbers 1, 4, 9, 16, … <br /> f3 = 9, f4= 16, f5 = 25, <br /> and a formula for fn is fn = n2.<br />b. For the sequence of even numbers 2, 4, 6, 8, … <br /> f3= 6, f4 = 8, f5 = 10, <br /> and a formula for fn is fn = 2*n <br />c. For the sequence of odd numbers 1, 3, 5, …<br /> a general formula is fn= 2n – 1.<br />
  17. 17. Sequences<br />Example B: <br />a. For the sequence of square numbers 1, 4, 9, 16, … <br /> f3 = 9, f4= 16, f5 = 25, <br /> and a formula for fn is fn = n2.<br />b. For the sequence of even numbers 2, 4, 6, 8, … <br /> f3= 6, f4 = 8, f5 = 10, <br /> and a formula for fn is fn = 2*n <br />c. For the sequence of odd numbers 1, 3, 5, …<br /> a general formula is fn= 2n – 1.<br />d. For the sequence of odd numbers with alternating <br /> signs -1, 3, -5, 7, -9, …fn = (-1)n(2n – 1)<br />
  18. 18. Sequences<br />Example B: <br />a. For the sequence of square numbers 1, 4, 9, 16, … <br /> f3 = 9, f4= 16, f5 = 25, <br /> and a formula for fn is fn = n2.<br />b. For the sequence of even numbers 2, 4, 6, 8, … <br /> f3= 6, f4 = 8, f5 = 10, <br /> and a formula for fn is fn = 2*n <br />c. For the sequence of odd numbers 1, 3, 5, …<br /> a general formula is fn= 2n – 1.<br />d. For the sequence of odd numbers with alternating <br /> signs -1, 3, -5, 7, -9, …fn = (-1)n(2n – 1)<br />A sequence whose signs alternate is called an alternating sequence as in part d. <br />
  19. 19. Summation Notation<br />
  20. 20. Summation Notation<br />In mathematics, the Greek letter “” (sigma) means <br />“to add”. <br />
  21. 21. Summation Notation<br />In mathematics, the Greek letter “” (sigma) means <br />“to add”. Hence, “x” means to add the x’s,<br />
  22. 22. Summation Notation<br />In mathematics, the Greek letter “” (sigma) means <br />“to add”. Hence, “x” means to add the x’s,<br />“(x*y)” means to add the x*y’s. <br />
  23. 23. Summation Notation<br />In mathematics, the Greek letter “” (sigma) means <br />“to add”. Hence, “x” means to add the x’s,<br />“(x*y)” means to add the x*y’s. (Of course the x's and xy's have to be given in the context.) <br />
  24. 24. Summation Notation<br />In mathematics, the Greek letter “” (sigma) means <br />“to add”. Hence, “x” means to add the x’s,<br />“(x*y)” means to add the x*y’s. (Of course the x's and xy's have to be given in the context.) <br />Given a list, let's say, of 100 numbers, f1, f2, f3,.., f100,<br />
  25. 25. Summation Notation<br />In mathematics, the Greek letter “” (sigma) means <br />“to add”. Hence, “x” means to add the x’s,<br />“(x*y)” means to add the x*y’s. (Of course the x's and xy's have to be given in the context.) <br />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:<br />100<br />fk<br />k = 1<br />
  26. 26. Summation Notation<br />In mathematics, the Greek letter “” (sigma) means <br />“to add”. Hence, “x” means to add the x’s,<br />“(x*y)” means to add the x*y’s. (Of course the x's and xy's have to be given in the context.) <br />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:<br />100<br />fk<br />k = 1<br />A variable which is called the <br />“index” variable, in this case k.<br />
  27. 27. Summation Notation<br />In mathematics, the Greek letter “” (sigma) means <br />“to add”. Hence, “x” means to add the x’s,<br />“(x*y)” means to add the x*y’s. (Of course the x's and xy's have to be given in the context.) <br />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:<br />100<br />fk<br />k = 1<br />A variable which is called the <br />“index” variable, in this case k.<br />k begins with the bottom number<br />and counts up (runs) to the top number.<br />
  28. 28. Summation Notation<br />In mathematics, the Greek letter “” (sigma) means <br />“to add”. Hence, “x” means to add the x’s,<br />“(x*y)” means to add the x*y’s. (Of course the x's and xy's have to be given in the context.) <br />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:<br />100<br />fk<br />k = 1<br />The beginning number<br />A variable which is called the <br />“index” variable, in this case k.<br />k begins with the bottom number<br />and counts up (runs) to the top number.<br />
  29. 29. Summation Notation<br />In mathematics, the Greek letter “” (sigma) means <br />“to add”. Hence, “x” means to add the x’s,<br />“(x*y)” means to add the x*y’s. (Of course the x's and xy's have to be given in the context.) <br />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:<br />The ending number<br />100<br />fk<br />k = 1<br />The beginning number<br />A variable which is called the <br />“index” variable, in this case k.<br />k begins with the bottom number<br />and counts up (runs) to the top number.<br />
  30. 30. Summation Notation<br />In mathematics, the Greek letter “” (sigma) means <br />“to add”. Hence, “x” means to add the x’s,<br />“(x*y)” means to add the x*y’s. (Of course the x's and xy's have to be given in the context.) <br />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:<br />The ending number<br />100<br />fk= f1 f2 f3 … f99 f100<br />k = 1<br />The beginning number<br />A variable which is called the <br />“index” variable, in this case k.<br />k begins with the bottom number<br />and counts up (runs) to the top number.<br />
  31. 31. Summation Notation<br />In mathematics, the Greek letter “” (sigma) means <br />“to add”. Hence, “x” means to add the x’s,<br />“(x*y)” means to add the x*y’s. (Of course the x's and xy's have to be given in the context.) <br />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:<br />The ending number<br />100<br />fk= f1+f2+f3+ … + f99+ f100<br />k = 1<br />The beginning number<br />A variable which is called the <br />“index” variable, in this case k.<br />k begins with the bottom number<br />and counts up (runs) to the top number.<br />
  32. 32. Summation Notation<br />Example C:<br />8<br /> fk =<br />k=4<br />5<br /> ai = <br />i=2<br />9<br /> xjyj = <br />j=6<br />n+3<br /> aj = <br />j=n<br />
  33. 33. Summation Notation<br />Example C:<br />8<br /> fk = f4+f5+f6+f7+ f8<br />k=4<br />5<br /> ai = <br />i=2<br />9<br /> xjyj = <br />j=6<br />n+3<br /> aj = <br />j=n<br />
  34. 34. Summation Notation<br />Example C:<br />8<br /> fk = f4+f5+f6+f7+ f8<br />k=4<br />5<br /> ai = a2+a3+a4+a5<br />i=2<br />9<br /> xjyj = <br />j=6<br />n+3<br /> aj = <br />j=n<br />
  35. 35. Summation Notation<br />Example C:<br />8<br /> fk = f4+f5+f6+f7+ f8<br />k=4<br />5<br /> ai = a2+a3+a4+a5<br />i=2<br />9<br /> xjyj = x6y6+x7y7+x8y8+x9y9<br />j=6<br />n+3<br /> aj = <br />j=n<br />
  36. 36. Summation Notation<br />Example C:<br />8<br /> fk = f4+f5+f6+f7+ f8<br />k=4<br />5<br /> ai = a2+a3+a4+a5<br />i=2<br />9<br /> xjyj = x6y6+x7y7+x8y8+x9y9<br />j=6<br />n+3<br /> aj = an+an+1+an+2+an+3<br />j=n<br />
  37. 37. Summation Notation<br />Example C:<br />8<br /> fk = f4+f5+f6+f7+ f8<br />k=4<br />5<br /> ai = a2+a3+a4+a5<br />i=2<br />9<br /> xjyj = x6y6+x7y7+x8y8+x9y9<br />j=6<br />n+3<br /> aj = an+an+1+an+2+an+3<br />j=n<br />Summation notation are used to express formulas in mathematics. <br />
  38. 38. Summation Notation<br />Example C:<br />8<br /> fk = f4+f5+f6+f7+ f8<br />k=4<br />5<br /> ai = a2+a3+a4+a5<br />i=2<br />9<br /> xjyj = x6y6+x7y7+x8y8+x9y9<br />j=6<br />n+3<br /> aj = an+an+1+an+2+an+3<br />j=n<br />Summation notation are used to express formulas in mathematics. An example is the formula for averaging. <br />
  39. 39. Summation Notation<br />Example C:<br />8<br /> fk = f4+f5+f6+f7+ f8<br />k=4<br />5<br /> ai = a2+a3+a4+a5<br />i=2<br />9<br /> xjyj = x6y6+x7y7+x8y8+x9y9<br />j=6<br />n+3<br /> aj = an+an+1+an+2+an+3<br />j=n<br />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, <br />is (f1 + f2 + f3 ... + fn-1 + fn)/n.<br />
  40. 40. Summation Notation<br />Example C:<br />8<br /> fk = f4+f5+f6+f7+ f8<br />k=4<br />5<br /> ai = a2+a3+a4+a5<br />i=2<br />9<br /> xjyj = x6y6+x7y7+x8y8+x9y9<br />j=6<br />n+3<br /> aj = an+an+1+an+2+an+3<br />j=n<br />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, <br />is (f1 + f2 + f3 ... + fn-1 + fn)/n.In notation,<br />n<br /><br /> fk<br />k=1<br /> f = <br />n<br />
  41. 41. Summation Notation<br />The index variable is also used as the variable that generates the numbers to be summed. <br />
  42. 42. Summation Notation<br />The index variable is also used as the variable that generates the numbers to be summed. <br />Example D:<br />8<br />a.  (k2 – 1) <br />k=5<br />
  43. 43. Summation Notation<br />The index variable is also used as the variable that generates the numbers to be summed. <br />Example D:<br />8<br />a.  (k2 – 1) = <br />k=5<br />k=5 k=6 k=7 k=8<br />
  44. 44. Summation Notation<br />The index variable is also used as the variable that generates the numbers to be summed. <br />Example D:<br />8<br />a.  (k2 – 1) = (52–1) + (62–1) + (72–1) + (82–1)<br />k=5<br />k=5 k=6 k=7 k=8<br />
  45. 45. Summation Notation<br />The index variable is also used as the variable that generates the numbers to be summed. <br />Example D:<br />8<br />a.  (k2 – 1) = (52–1) + (62–1) + (72–1) + (82–1)<br /> = 24 + 35 + 48 + 63<br />k=5<br />k=5 k=6 k=7 k=8<br />
  46. 46. Summation Notation<br />The index variable is also used as the variable that generates the numbers to be summed. <br />Example D:<br />8<br />a.  (k2 – 1) = (52–1) + (62–1) + (72–1) + (82–1)<br /> = 24 + 35 + 48 + 63<br /> = 170 <br />k=5<br />k=5 k=6 k=7 k=8<br />
  47. 47. Summation Notation<br />The index variable is also used as the variable that generates the numbers to be summed. <br />Example D:<br />8<br />a.  (k2 – 1) = (52–1) + (62–1) + (72–1) + (82–1)<br /> = 24 + 35 + 48 + 63<br /> = 170 <br />k=5<br />k=5 k=6 k=7 k=8<br />5<br />b.  (-1)k(3k + 2)<br />k=3 <br />
  48. 48. Summation Notation<br />The index variable is also used as the variable that generates the numbers to be summed. <br />Example D:<br />8<br />a.  (k2 – 1) = (52–1) + (62–1) + (72–1) + (82–1)<br /> = 24 + 35 + 48 + 63<br /> = 170 <br />k=5<br />k=5 k=6 k=7 k=8<br />5<br />b.  (-1)k(3k + 2)<br /> =(-1)3(3*3+2)+(-1)4(3*4+2)+(-1)5(3*5+2)<br />k=3 <br />
  49. 49. Summation Notation<br />The index variable is also used as the variable that generates the numbers to be summed. <br />Example D:<br />8<br />a.  (k2 – 1) = (52–1) + (62–1) + (72–1) + (82–1)<br /> = 24 + 35 + 48 + 63<br /> = 170 <br />k=5<br />k=5 k=6 k=7 k=8<br />5<br />b.  (-1)k(3k + 2)<br /> =(-1)3(3*3+2)+(-1)4(3*4+2)+(-1)5(3*5+2)<br /> = -11 + 14 – 17<br />k=3 <br />
  50. 50. Summation Notation<br />The index variable is also used as the variable that generates the numbers to be summed. <br />Example D:<br />8<br />a.  (k2 – 1) = (52–1) + (62–1) + (72–1) + (82–1)<br /> = 24 + 35 + 48 + 63<br /> = 170 <br />k=5<br />k=5 k=6 k=7 k=8<br />5<br />b.  (-1)k(3k + 2)<br /> =(-1)3(3*3+2)+(-1)4(3*4+2)+(-1)5(3*5+2)<br /> = -11 + 14 – 17<br /> = -14 <br />k=3 <br />
  51. 51. Summation Notation<br />The index variable is also used as the variable that generates the numbers to be summed. <br />Example D:<br />8<br />a.  (k2 – 1) = (52–1) + (62–1) + (72–1) + (82–1)<br /> = 24 + 35 + 48 + 63<br /> = 170 <br />k=5<br />k=5 k=6 k=7 k=8<br />5<br />b.  (-1)k(3k + 2)<br /> =(-1)3(3*3+2)+(-1)4(3*4+2)+(-1)5(3*5+2)<br /> = -11 + 14 – 17<br /> = -14 <br />k=3 <br />In part b, the multiple (-1)k change the sums to an alternating sum, t<br />
  52. 52. Summation Notation<br />The index variable is also used as the variable that generates the numbers to be summed. <br />Example D:<br />8<br />a.  (k2 – 1) = (52–1) + (62–1) + (72–1) + (82–1)<br /> = 24 + 35 + 48 + 63<br /> = 170 <br />k=5<br />k=5 k=6 k=7 k=8<br />5<br />b.  (-1)k(3k + 2)<br /> =(-1)3(3*3+2)+(-1)4(3*4+2)+(-1)5(3*5+2)<br /> = -11 + 14 – 17<br /> = -14 <br />k=3 <br />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. <br />
  53. 53. Properties Summation Notation<br />a.  (ak + bk) = <br /> ak +<br /> bk <br />k <br />k <br />k <br />
  54. 54. Properties Summation Notation<br />a.  (ak + bk) = <br /> ak +<br /> bk <br />k <br />k <br />k <br />b.  (ak – bk) = <br /> ak – <br /> bk <br />k <br />k <br />k <br />
  55. 55. Properties Summation Notation<br />a.  (ak + bk) = <br /> ak +<br /> bk <br />k <br />k <br />k <br />b.  (ak – bk) = <br /> ak – <br /> bk <br />k <br />k <br />k <br />c.  cak = <br />c(ak) where c is a constant.<br />k <br />k <br />
  56. 56. Properties Summation Notation<br />a.  (ak + bk) = <br /> ak +<br /> bk <br />k <br />k <br />k <br />b.  (ak – bk) = <br /> ak – <br /> bk <br />k <br />k <br />k <br />c.  cak = <br />c(ak) where c is a constant.<br />k <br />k <br />d. If c is a constant, then c + c + .. + c = nc,<br /> hence c = nc<br />n<br />n times<br />k=1 <br />
  57. 57. Properties Summation Notation<br />a.  (ak + bk) = <br /> ak +<br /> bk <br />k <br />k <br />k <br />b.  (ak – bk) = <br /> ak – <br /> bk <br />k <br />k <br />k <br />c.  cak = <br />c(ak) where c is a constant.<br />k <br />k <br />d. If c is a constant, then c + c + .. + c = nc,<br /> hence c = nc<br />n<br />n times<br />k=1 <br />To see part a, write out the terms in the summations.<br />
  58. 58. Properties Summation Notation<br />a.  (ak + bk) = <br /> ak +<br /> bk <br />k <br />k <br />k <br />b.  (ak – bk) = <br /> ak – <br /> bk <br />k <br />k <br />k <br />c.  cak = <br />c(ak) where c is a constant.<br />k <br />k <br />d. If c is a constant, then c + c + .. + c = nc,<br /> hence c = nc<br />n<br />n times<br />k=1 <br />To see part a, write out the terms in the summations.<br />n <br /> (ak + bk) = <br />k=1 <br />
  59. 59. Properties Summation Notation<br />a.  (ak + bk) = <br /> ak +<br /> bk <br />k <br />k <br />k <br />b.  (ak – bk) = <br /> ak – <br /> bk <br />k <br />k <br />k <br />c.  cak = <br />c(ak) where c is a constant.<br />k <br />k <br />d. If c is a constant, then c + c + .. + c = nc,<br /> hence c = nc<br />n<br />n times<br />k=1 <br />To see part a, write out the terms in the summations.<br />n <br /> (ak + bk) = (a1 + b1) + (a2 + b2) + .. + (an + bn)<br />k=1 <br />
  60. 60. Properties Summation Notation<br />a.  (ak + bk) = <br /> ak +<br /> bk <br />k <br />k <br />k <br />b.  (ak – bk) = <br /> ak – <br /> bk <br />k <br />k <br />k <br />c.  cak = <br />c(ak) where c is a constant.<br />k <br />k <br />d. If c is a constant, then c + c + .. + c = nc,<br /> hence c = nc<br />n<br />n times<br />k=1 <br />To see part a, write out the terms in the summations.<br />n <br /> (ak + bk) = (a1 + b1) + (a2 + b2) + .. + (an + bn)<br /> = (a1 + a2 + .. + an) + (b1 + b2 + .. + bn)<br />k=1 <br />
  61. 61. Properties Summation Notation<br />a.  (ak + bk) = <br /> ak +<br /> bk <br />k <br />k <br />k <br />b.  (ak – bk) = <br /> ak – <br /> bk <br />k <br />k <br />k <br />c.  cak = <br />c(ak) where c is a constant.<br />k <br />k <br />d. If c is a constant, then c + c + .. + c = nc,<br /> hence c = nc<br />n<br />n times<br />k=1 <br />To see part a, write out the terms in the summations.<br />n <br /> (ak + bk) = (a1 + b1) + (a2 + b2) + .. + (an + bn)<br /> = (a1 + a2 + .. + an) + (b1 + b2 + .. + bn)<br /> = <br />k=1 <br /> ak +<br /> bk <br />k <br />k <br />
  62. 62. Properties Summation Notation<br />a.  (ak + bk) = <br /> ak +<br /> bk <br />k <br />k <br />k <br />b.  (ak – bk) = <br /> ak – <br /> bk <br />k <br />k <br />k <br />c.  cak = <br />c(ak) where c is a constant.<br />k <br />k <br />d. If c is a constant, then c + c + .. + c = nc,<br /> hence c = nc<br />n<br />n times<br />k=1 <br />To see part a, write out the terms in the summations.<br />n <br /> (ak + bk) = (a1 + b1) + (a2 + b2) + .. + (an + bn)<br /> = (a1 + a2 + .. + an) + (b1 + b2 + .. + bn)<br /> = <br />k=1 <br /> ak +<br /> bk <br />k <br />k <br />The other parts may be verified similarly.<br />
  63. 63. Properties Summation Notation<br />Let S = 1 + 2 + .. + (n – 1) + n, <br />
  64. 64. Properties Summation Notation<br />Let S = 1 + 2 + .. + (n – 1) + n, take two copies of the sums and sum them in the following manner: <br />
  65. 65. Properties Summation Notation<br />Let S = 1 + 2 + .. + (n – 1) + n, take two copies of the sums and sum them in the following manner: <br />S = 1 + 2 + ……+ (n – 1) + n<br />S = n + (n – 1) + ……+ 2 + 1 <br />
  66. 66. Properties Summation Notation<br />Let S = 1 + 2 + .. + (n – 1) + n, take two copies of the sums and sum them in the following manner: <br />S = 1 + 2 + ……+ (n – 1) + n<br />S = n + (n – 1) + ……+ 2 + 1 <br />2S=<br />
  67. 67. Properties Summation Notation<br />Let S = 1 + 2 + .. + (n – 1) + n, take two copies of the sums and sum them in the following manner: <br />S = 1 + 2 + ……+ (n – 1) + n<br />S = n + (n – 1) + ……+ 2 + 1 <br />2S=(n+1)<br />
  68. 68. Properties Summation Notation<br />Let S = 1 + 2 + .. + (n – 1) + n, take two copies of the sums and sum them in the following manner: <br />S = 1 + 2 + ……+ (n – 1) + n<br />S = n + (n – 1) + ……+ 2 + 1 <br />2S=(n+1)+(n+1)<br />
  69. 69. Properties Summation Notation<br />Let S = 1 + 2 + .. + (n – 1) + n, take two copies of the sums and sum them in the following manner: <br />S = 1 + 2 + ……+ (n – 1) + n<br />S = n + (n – 1) + ……+ 2 + 1 <br />2S=(n+1)+(n+1)+……+(n+1)+(n+1) <br />
  70. 70. Properties Summation Notation<br />Let S = 1 + 2 + .. + (n – 1) + n, take two copies of the sums and sum them in the following manner: <br />S = 1 + 2 + ……+ (n – 1) + n<br />S = n + (n – 1) + ……+ 2 + 1 <br />2S=(n+1)+(n+1)+……+(n+1)+(n+1) <br />n times<br />
  71. 71. Properties Summation Notation<br />Let S = 1 + 2 + .. + (n – 1) + n, take two copies of the sums and sum them in the following manner: <br />S = 1 + 2 + ……+ (n – 1) + n<br />S = n + (n – 1) + ……+ 2 + 1 <br />2S=(n+1)+(n+1)+……+(n+1)+(n+1) <br />n times<br />Hence 2S = n(n + 1) <br />
  72. 72. Properties Summation Notation<br />Let S = 1 + 2 + .. + (n – 1) + n, take two copies of the sums and sum them in the following manner: <br />S = 1 + 2 + ……+ (n – 1) + n<br />S = n + (n – 1) + ……+ 2 + 1 <br />2S=(n+1)+(n+1)+……+(n+1)+(n+1) <br />n times<br />n(n + 1)<br />Hence 2S = n(n + 1) or S = <br />2<br />
  73. 73. Properties Summation Notation<br />Let S = 1 + 2 + .. + (n – 1) + n, take two copies of the sums and sum them in the following manner: <br />S = 1 + 2 + ……+ (n – 1) + n<br />S = n + (n – 1) + ……+ 2 + 1 <br />2S=(n+1)+(n+1)+……+(n+1)+(n+1) <br />n times<br />n(n + 1)<br />Hence 2S = n(n + 1) or S = <br />2<br />Formula for the Sum of Natural Numbers:<br />n(n + 1)<br />n <br /> k = <br />2<br />k=1 <br />
  74. 74. Properties Summation Notation<br />Let S = 1 + 2 + .. + (n – 1) + n, take two copies of the sums and sum them in the following manner: <br />S = 1 + 2 + ……+ (n – 1) + n<br />S = n + (n – 1) + ……+ 2 + 1 <br />2S=(n+1)+(n+1)+……+(n+1)+(n+1) <br />n times<br />n(n + 1)<br />Hence 2S = n(n + 1) or S = <br />2<br />Formula for the Sum of Natural Numbers:<br />n(n + 1)<br />n <br /> k = <br />2<br />k=1 <br />100<br />For example 1+2+3..+100 =  k<br />k=1<br />
  75. 75. Properties Summation Notation<br />Let S = 1 + 2 + .. + (n – 1) + n, take two copies of the sums and sum them in the following manner: <br />S = 1 + 2 + ……+ (n – 1) + n<br />S = n + (n – 1) + ……+ 2 + 1 <br />2S=(n+1)+(n+1)+……+(n+1)+(n+1) <br />n times<br />n(n + 1)<br />Hence 2S = n(n + 1) or S = <br />2<br />Formula for the Sum of Natural Numbers:<br />n(n + 1)<br />n <br /> k = <br />2<br />k=1 <br />100(100 + 1)<br />100<br />For example 1+2+3..+100 =  k=<br />2<br />k=1<br />= 5050<br />
  76. 76. Properties Summation Notation<br />We may use the above properties and the sum formula to sum all linear sums.<br />
  77. 77. Properties Summation Notation<br />We may use the above properties and the sum formula to sum all linear sums.<br />45 <br />Example E: Find  (2k – 5) <br />k=1 <br />
  78. 78. Properties Summation Notation<br />We may use the above properties and the sum formula to sum all linear sums.<br />45 <br />Example E: Find  (2k – 5) <br />k=1 <br />45 <br /> (2k – 5) = Σ2k – Σ5 by property a <br />k <br />k <br />k=1 <br />
  79. 79. Properties Summation Notation<br />We may use the above properties and the sum formula to sum all linear sums.<br />45 <br />Example E: Find  (2k – 5) <br />k=1 <br />45 <br /> (2k – 5) = Σ2k – Σ5 by property a <br />k <br />k <br />k=1 <br />45 <br />45 <br /> = 2Σk – Σ5 by property c <br />k=1 <br />k=1 <br />
  80. 80. Properties Summation Notation<br />We may use the above properties and the sum formula to sum all linear sums.<br />45 <br />Example E: Find  (2k – 5) <br />k=1 <br />45 <br /> (2k – 5) = Σ2k – Σ5 by property a <br />k <br />k <br />k=1 <br />45 <br />45 <br /> = 2Σk – Σ5 by property c <br />k=1 <br />k=1 <br />45(45 + 1)<br /> = 2 – 5*45 <br />2<br />
  81. 81. Properties Summation Notation<br />We may use the above properties and the sum formula to sum all linear sums.<br />45 <br />Example E: Find  (2k – 5) <br />k=1 <br />45 <br /> (2k – 5) = Σ2k – Σ5 by property a <br />k <br />k <br />k=1 <br />45 <br />45 <br /> = 2Σk – Σ5 by property c <br />k=1 <br />k=1 <br />45(45 + 1)<br /> = 2 – 5*45 <br />2<br />by property d<br />by the sum formula<br />
  82. 82. Properties Summation Notation<br />We may use the above properties and the sum formula to sum all linear sums.<br />45 <br />Example E: Find  (2k – 5) <br />k=1 <br />45 <br /> (2k – 5) = Σ2k – Σ5 by property a <br />k <br />k <br />k=1 <br />45 <br />45 <br /> = 2Σk – Σ5 by property c <br />k=1 <br />k=1 <br />45(45 + 1)<br /> = 2 – 5*45 <br />2<br />by property d<br />by the sum formula<br /> = 2070 – 225 = 1845 <br />
  83. 83. Properties Summation Notation<br />The index variable may be shifted in order to utilize the sum formula.<br />
  84. 84. Properties Summation Notation<br />The index variable may be shifted in order to utilize the sum formula.<br />53 <br />Example E: Find  (4k – 33) <br />k=10 <br />
  85. 85. Properties Summation Notation<br />The index variable may be shifted in order to utilize the sum formula.<br />53 <br />Example F: Find  (4k – 33) <br />k=10 <br />Select a new index, say m, to start at 1.<br />
  86. 86. Properties Summation Notation<br />The index variable may be shifted in order to utilize the sum formula.<br />53 <br />Example F: Find  (4k – 33) <br />k=10 <br />Select a new index, say m, to start at 1.<br />The lower numbers are k = 10 and m = 1  k = m + 9.<br />
  87. 87. Properties Summation Notation<br />The index variable may be shifted in order to utilize the sum formula.<br />53 <br />Example F: Find  (4k – 33) <br />k=10 <br />Select a new index, say m, to start at 1.<br />The lower numbers are k = 10 and m = 1  k = m + 9.<br />The upper number is k = 53  53 = m + 9 <br />
  88. 88. Properties Summation Notation<br />The index variable may be shifted in order to utilize the sum formula.<br />53 <br />Example F: Find  (4k – 33) <br />k=10 <br />Select a new index, say m, to start at 1.<br />The lower numbers are k = 10 and m = 1  k = m + 9.<br />The upper number is k = 53  53 = m + 9 or m = 44. <br />
  89. 89. Properties Summation Notation<br />The index variable may be shifted in order to utilize the sum formula.<br />53 <br />Example F: Find  (4k – 33) <br />k=10 <br />Select a new index, say m, to start at 1.<br />The lower numbers are k = 10 and m = 1  k = m + 9.<br />The upper number is k = 53  53 = m + 9 or m = 44. <br />Rewrite the sum in terms of m: <br />
  90. 90. Properties Summation Notation<br />The index variable may be shifted in order to utilize the sum formula.<br />53 <br />Example F: Find  (4k – 33) <br />k=10 <br />Select a new index, say m, to start at 1.<br />The lower numbers are k = 10 and m = 1  k = m + 9.<br />The upper number is k = 53  53 = m + 9 or m = 44. <br />Rewrite the sum in terms of m: <br />53 <br />44<br /> (4k – 33) = Σ[4(m + 9) – 33] <br />m=1<br />k=10 <br />
  91. 91. Properties Summation Notation<br />The index variable may be shifted in order to utilize the sum formula.<br />53 <br />Example F: Find  (4k – 33) <br />k=10 <br />Select a new index, say m, to start at 1.<br />The lower numbers are k = 10 and m = 1  k = m + 9.<br />The upper number is k = 53  53 = m + 9 or m = 44. <br />Rewrite the sum in terms of m: <br />53 <br />44<br />44<br /> (4k – 33) = Σ[4(m + 9) – 33] = Σ(4m – 3) <br />m=1<br />m=1<br />k=10 <br />
  92. 92. Properties Summation Notation<br />The index variable may be shifted in order to utilize the sum formula.<br />53 <br />Example F: Find  (4k – 33) <br />k=10 <br />Select a new index, say m, to start at 1.<br />The lower numbers are k = 10 and m = 1  k = m + 9.<br />The upper number is k = 53  53 = m + 9 or m = 44. <br />Rewrite the sum in terms of m: <br />53 <br />44<br />44<br /> (4k – 33) = Σ[4(m + 9) – 33] = Σ(4m – 3) <br />m=1<br />m=1<br />k=10 <br />44<br />44<br /> = 4Σm – Σ3 <br />m=1<br />m=1<br />
  93. 93. Properties Summation Notation<br />The index variable may be shifted in order to utilize the sum formula.<br />53 <br />Example F: Find  (4k – 33) <br />k=10 <br />Select a new index, say m, to start at 1.<br />The lower numbers are k = 10 and m = 1  k = m + 9.<br />The upper number is k = 53  53 = m + 9 or m = 44. <br />Rewrite the sum in terms of m: <br />53 <br />44<br />44<br /> (4k – 33) = Σ[4(m + 9) – 33] = Σ(4m – 3) <br />m=1<br />m=1<br />k=10 <br />44<br />44<br /> = 4Σm – Σ3 <br />m=1<br />m=1<br />44(44 + 1)<br /> = 4 – 3*44 <br />2<br /> = 3828 <br />

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