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5.1 sequences and summation notation
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5.1 sequences and summation notation

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    5.1 sequences and summation notation 5.1 sequences and summation notation Presentation Transcript

    • Sequences
    • Sequences
      A sequence is an ordered list of infinitely many numbers that may or may not have a pattern.
    • Sequences
      A sequence is an ordered list of infinitely many numbers that may or may not have a pattern.
      Example A:
      1, 3, 5, 7, 9,… is the sequence of odd numbers.
    • Sequences
      A sequence is an ordered list of infinitely many numbers that may or may not have a pattern.
      Example A:
      1, 3, 5, 7, 9,… is the sequence of odd numbers.
      1, 4, 9, 16, 25,… is the sequence of square numbers.
    • Sequences
      A sequence is an ordered list of infinitely many numbers that may or may not have a pattern.
      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.
    • Sequences
      A sequence is an ordered list of infinitely many numbers that may or may not have a pattern.
      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.
      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
      A sequence is an ordered list of infinitely many numbers that may or may not have a pattern.
      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.
      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.
      A sequence may be listed as f1, f2 , f3 , …
    • Sequences
      A sequence is an ordered list of infinitely many numbers that may or may not have a pattern.
      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.
      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.
      A sequence may be listed as f1, f2 , f3 , …
      f100 = 100th number on the list,
    • Sequences
      A sequence is an ordered list of infinitely many numbers that may or may not have a pattern.
      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.
      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.
      A sequence may be listed as f1, f2 , f3 , …
      f100 = 100th number on the list,
      fn = the n’th number on the list,
    • Sequences
      A sequence is an ordered list of infinitely many numbers that may or may not have a pattern.
      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.
      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.
      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.
    • Sequences
      Example B:
      a. For the sequence of square numbers 1, 4, 9, 16, …
      f3 = 9, f4= 16, f5 = 25,
    • Sequences
      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
      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
      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
      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
      c. For the sequence of odd numbers 1, 3, 5, …
      a general formula is fn= 2n – 1.
    • Sequences
      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
      c. For the sequence of odd numbers 1, 3, 5, …
      a general formula is fn= 2n – 1.
    • Sequences
      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
      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)
    • Sequences
      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
      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 in part d.
    • Summation Notation
    • Summation Notation
      In mathematics, the Greek letter “” (sigma) means
      “to add”.
    • Summation Notation
      In mathematics, the Greek letter “” (sigma) means
      “to add”. Hence, “x” means to add the x’s,
    • Summation Notation
      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
      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
      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.)
      Given a list, let's say, of 100 numbers, f1, f2, f3,.., f100,
    • Summation Notation
      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.)
      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:
      100
      fk
      k = 1
    • Summation Notation
      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.)
      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:
      100
      fk
      k = 1
      A variable which is called the
      “index” variable, in this case k.
    • Summation Notation
      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.)
      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:
      100
      fk
      k = 1
      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.
    • Summation Notation
      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.)
      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:
      100
      fk
      k = 1
      The beginning number
      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.
    • Summation Notation
      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.)
      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:
      The ending number
      100
      fk
      k = 1
      The beginning number
      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.
    • Summation Notation
      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.)
      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:
      The ending number
      100
      fk= f1 f2 f3 … f99 f100
      k = 1
      The beginning number
      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.
    • Summation Notation
      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.)
      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:
      The ending number
      100
      fk= f1+f2+f3+ … + f99+ f100
      k = 1
      The beginning number
      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.
    • Summation Notation
      Example C:
      8
       fk =
      k=4
      5
       ai =
      i=2
      9
       xjyj =
      j=6
      n+3
       aj =
      j=n
    • Summation Notation
      Example C:
      8
       fk = f4+f5+f6+f7+ f8
      k=4
      5
       ai =
      i=2
      9
       xjyj =
      j=6
      n+3
       aj =
      j=n
    • Summation Notation
      Example C:
      8
       fk = f4+f5+f6+f7+ f8
      k=4
      5
       ai = a2+a3+a4+a5
      i=2
      9
       xjyj =
      j=6
      n+3
       aj =
      j=n
    • Summation Notation
      Example C:
      8
       fk = f4+f5+f6+f7+ f8
      k=4
      5
       ai = a2+a3+a4+a5
      i=2
      9
       xjyj = x6y6+x7y7+x8y8+x9y9
      j=6
      n+3
       aj =
      j=n
    • Summation Notation
      Example C:
      8
       fk = f4+f5+f6+f7+ f8
      k=4
      5
       ai = a2+a3+a4+a5
      i=2
      9
       xjyj = x6y6+x7y7+x8y8+x9y9
      j=6
      n+3
       aj = an+an+1+an+2+an+3
      j=n
    • Summation Notation
      Example C:
      8
       fk = f4+f5+f6+f7+ f8
      k=4
      5
       ai = a2+a3+a4+a5
      i=2
      9
       xjyj = x6y6+x7y7+x8y8+x9y9
      j=6
      n+3
       aj = an+an+1+an+2+an+3
      j=n
      Summation notation are used to express formulas in mathematics.
    • Summation Notation
      Example C:
      8
       fk = f4+f5+f6+f7+ f8
      k=4
      5
       ai = a2+a3+a4+a5
      i=2
      9
       xjyj = x6y6+x7y7+x8y8+x9y9
      j=6
      n+3
       aj = an+an+1+an+2+an+3
      j=n
      Summation notation are used to express formulas in mathematics. An example is the formula for averaging.
    • Summation Notation
      Example C:
      8
       fk = f4+f5+f6+f7+ f8
      k=4
      5
       ai = a2+a3+a4+a5
      i=2
      9
       xjyj = x6y6+x7y7+x8y8+x9y9
      j=6
      n+3
       aj = an+an+1+an+2+an+3
      j=n
      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.
    • Summation Notation
      Example C:
      8
       fk = f4+f5+f6+f7+ f8
      k=4
      5
       ai = a2+a3+a4+a5
      i=2
      9
       xjyj = x6y6+x7y7+x8y8+x9y9
      j=6
      n+3
       aj = an+an+1+an+2+an+3
      j=n
      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,
      n

      fk
      k=1
      f =
      n
    • Summation Notation
      The index variable is also used as the variable that generates the numbers to be summed.
    • Summation Notation
      The index variable is also used as the variable that generates the numbers to be summed.
      Example D:
      8
      a.  (k2 – 1)
      k=5
    • Summation Notation
      The index variable is also used as the variable that generates the numbers to be summed.
      Example D:
      8
      a.  (k2 – 1) =
      k=5
      k=5 k=6 k=7 k=8
    • Summation Notation
      The index variable is also used as the variable that generates the numbers to be summed.
      Example D:
      8
      a.  (k2 – 1) = (52–1) + (62–1) + (72–1) + (82–1)
      k=5
      k=5 k=6 k=7 k=8
    • Summation Notation
      The index variable is also used as the variable that generates the numbers to be summed.
      Example D:
      8
      a.  (k2 – 1) = (52–1) + (62–1) + (72–1) + (82–1)
      = 24 + 35 + 48 + 63
      k=5
      k=5 k=6 k=7 k=8
    • Summation Notation
      The index variable is also used as the variable that generates the numbers to be summed.
      Example D:
      8
      a.  (k2 – 1) = (52–1) + (62–1) + (72–1) + (82–1)
      = 24 + 35 + 48 + 63
      = 170
      k=5
      k=5 k=6 k=7 k=8
    • Summation Notation
      The index variable is also used as the variable that generates the numbers to be summed.
      Example D:
      8
      a.  (k2 – 1) = (52–1) + (62–1) + (72–1) + (82–1)
      = 24 + 35 + 48 + 63
      = 170
      k=5
      k=5 k=6 k=7 k=8
      5
      b.  (-1)k(3k + 2)
      k=3
    • Summation Notation
      The index variable is also used as the variable that generates the numbers to be summed.
      Example D:
      8
      a.  (k2 – 1) = (52–1) + (62–1) + (72–1) + (82–1)
      = 24 + 35 + 48 + 63
      = 170
      k=5
      k=5 k=6 k=7 k=8
      5
      b.  (-1)k(3k + 2)
      =(-1)3(3*3+2)+(-1)4(3*4+2)+(-1)5(3*5+2)
      k=3
    • Summation Notation
      The index variable is also used as the variable that generates the numbers to be summed.
      Example D:
      8
      a.  (k2 – 1) = (52–1) + (62–1) + (72–1) + (82–1)
      = 24 + 35 + 48 + 63
      = 170
      k=5
      k=5 k=6 k=7 k=8
      5
      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
    • Summation Notation
      The index variable is also used as the variable that generates the numbers to be summed.
      Example D:
      8
      a.  (k2 – 1) = (52–1) + (62–1) + (72–1) + (82–1)
      = 24 + 35 + 48 + 63
      = 170
      k=5
      k=5 k=6 k=7 k=8
      5
      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
    • Summation Notation
      The index variable is also used as the variable that generates the numbers to be summed.
      Example D:
      8
      a.  (k2 – 1) = (52–1) + (62–1) + (72–1) + (82–1)
      = 24 + 35 + 48 + 63
      = 170
      k=5
      k=5 k=6 k=7 k=8
      5
      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
      In part b, the multiple (-1)k change the sums to an alternating sum, t
    • Summation Notation
      The index variable is also used as the variable that generates the numbers to be summed.
      Example D:
      8
      a.  (k2 – 1) = (52–1) + (62–1) + (72–1) + (82–1)
      = 24 + 35 + 48 + 63
      = 170
      k=5
      k=5 k=6 k=7 k=8
      5
      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
      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.
    • Properties Summation Notation
      a.  (ak + bk) =
       ak +
       bk
      k
      k
      k
    • Properties Summation Notation
      a.  (ak + bk) =
       ak +
       bk
      k
      k
      k
      b.  (ak – bk) =
       ak –
       bk
      k
      k
      k
    • Properties Summation Notation
      a.  (ak + bk) =
       ak +
       bk
      k
      k
      k
      b.  (ak – bk) =
       ak –
       bk
      k
      k
      k
      c.  cak =
      c(ak) where c is a constant.
      k
      k
    • Properties Summation Notation
      a.  (ak + bk) =
       ak +
       bk
      k
      k
      k
      b.  (ak – bk) =
       ak –
       bk
      k
      k
      k
      c.  cak =
      c(ak) where c is a constant.
      k
      k
      d. If c is a constant, then c + c + .. + c = nc,
      hence c = nc
      n
      n times
      k=1
    • Properties Summation Notation
      a.  (ak + bk) =
       ak +
       bk
      k
      k
      k
      b.  (ak – bk) =
       ak –
       bk
      k
      k
      k
      c.  cak =
      c(ak) where c is a constant.
      k
      k
      d. If c is a constant, then c + c + .. + c = nc,
      hence c = nc
      n
      n times
      k=1
      To see part a, write out the terms in the summations.
    • Properties Summation Notation
      a.  (ak + bk) =
       ak +
       bk
      k
      k
      k
      b.  (ak – bk) =
       ak –
       bk
      k
      k
      k
      c.  cak =
      c(ak) where c is a constant.
      k
      k
      d. If c is a constant, then c + c + .. + c = nc,
      hence c = nc
      n
      n times
      k=1
      To see part a, write out the terms in the summations.
      n
       (ak + bk) =
      k=1
    • Properties Summation Notation
      a.  (ak + bk) =
       ak +
       bk
      k
      k
      k
      b.  (ak – bk) =
       ak –
       bk
      k
      k
      k
      c.  cak =
      c(ak) where c is a constant.
      k
      k
      d. If c is a constant, then c + c + .. + c = nc,
      hence c = nc
      n
      n times
      k=1
      To see part a, write out the terms in the summations.
      n
       (ak + bk) = (a1 + b1) + (a2 + b2) + .. + (an + bn)
      k=1
    • Properties Summation Notation
      a.  (ak + bk) =
       ak +
       bk
      k
      k
      k
      b.  (ak – bk) =
       ak –
       bk
      k
      k
      k
      c.  cak =
      c(ak) where c is a constant.
      k
      k
      d. If c is a constant, then c + c + .. + c = nc,
      hence c = nc
      n
      n times
      k=1
      To see part a, write out the terms in the summations.
      n
       (ak + bk) = (a1 + b1) + (a2 + b2) + .. + (an + bn)
      = (a1 + a2 + .. + an) + (b1 + b2 + .. + bn)
      k=1
    • Properties Summation Notation
      a.  (ak + bk) =
       ak +
       bk
      k
      k
      k
      b.  (ak – bk) =
       ak –
       bk
      k
      k
      k
      c.  cak =
      c(ak) where c is a constant.
      k
      k
      d. If c is a constant, then c + c + .. + c = nc,
      hence c = nc
      n
      n times
      k=1
      To see part a, write out the terms in the summations.
      n
       (ak + bk) = (a1 + b1) + (a2 + b2) + .. + (an + bn)
      = (a1 + a2 + .. + an) + (b1 + b2 + .. + bn)
      =
      k=1
       ak +
       bk
      k
      k
    • Properties Summation Notation
      a.  (ak + bk) =
       ak +
       bk
      k
      k
      k
      b.  (ak – bk) =
       ak –
       bk
      k
      k
      k
      c.  cak =
      c(ak) where c is a constant.
      k
      k
      d. If c is a constant, then c + c + .. + c = nc,
      hence c = nc
      n
      n times
      k=1
      To see part a, write out the terms in the summations.
      n
       (ak + bk) = (a1 + b1) + (a2 + b2) + .. + (an + bn)
      = (a1 + a2 + .. + an) + (b1 + b2 + .. + bn)
      =
      k=1
       ak +
       bk
      k
      k
      The other parts may be verified similarly.
    • Properties Summation Notation
      Let S = 1 + 2 + .. + (n – 1) + n,
    • Properties Summation Notation
      Let S = 1 + 2 + .. + (n – 1) + n, take two copies of the sums and sum them in the following manner:
    • Properties Summation Notation
      Let S = 1 + 2 + .. + (n – 1) + n, take two copies of the sums and sum them in the following manner:
      S = 1 + 2 + ……+ (n – 1) + n
      S = n + (n – 1) + ……+ 2 + 1
    • Properties Summation Notation
      Let S = 1 + 2 + .. + (n – 1) + n, take two copies of the sums and sum them in the following manner:
      S = 1 + 2 + ……+ (n – 1) + n
      S = n + (n – 1) + ……+ 2 + 1
      2S=
    • Properties Summation Notation
      Let S = 1 + 2 + .. + (n – 1) + n, take two copies of the sums and sum them in the following manner:
      S = 1 + 2 + ……+ (n – 1) + n
      S = n + (n – 1) + ……+ 2 + 1
      2S=(n+1)
    • Properties Summation Notation
      Let S = 1 + 2 + .. + (n – 1) + n, take two copies of the sums and sum them in the following manner:
      S = 1 + 2 + ……+ (n – 1) + n
      S = n + (n – 1) + ……+ 2 + 1
      2S=(n+1)+(n+1)
    • Properties Summation Notation
      Let S = 1 + 2 + .. + (n – 1) + n, take two copies of the sums and sum them in the following manner:
      S = 1 + 2 + ……+ (n – 1) + n
      S = n + (n – 1) + ……+ 2 + 1
      2S=(n+1)+(n+1)+……+(n+1)+(n+1)
    • Properties Summation Notation
      Let S = 1 + 2 + .. + (n – 1) + n, take two copies of the sums and sum them in the following manner:
      S = 1 + 2 + ……+ (n – 1) + n
      S = n + (n – 1) + ……+ 2 + 1
      2S=(n+1)+(n+1)+……+(n+1)+(n+1)
      n times
    • Properties Summation Notation
      Let S = 1 + 2 + .. + (n – 1) + n, take two copies of the sums and sum them in the following manner:
      S = 1 + 2 + ……+ (n – 1) + n
      S = n + (n – 1) + ……+ 2 + 1
      2S=(n+1)+(n+1)+……+(n+1)+(n+1)
      n times
      Hence 2S = n(n + 1)
    • Properties Summation Notation
      Let S = 1 + 2 + .. + (n – 1) + n, take two copies of the sums and sum them in the following manner:
      S = 1 + 2 + ……+ (n – 1) + n
      S = n + (n – 1) + ……+ 2 + 1
      2S=(n+1)+(n+1)+……+(n+1)+(n+1)
      n times
      n(n + 1)
      Hence 2S = n(n + 1) or S =
      2
    • Properties Summation Notation
      Let S = 1 + 2 + .. + (n – 1) + n, take two copies of the sums and sum them in the following manner:
      S = 1 + 2 + ……+ (n – 1) + n
      S = n + (n – 1) + ……+ 2 + 1
      2S=(n+1)+(n+1)+……+(n+1)+(n+1)
      n times
      n(n + 1)
      Hence 2S = n(n + 1) or S =
      2
      Formula for the Sum of Natural Numbers:
      n(n + 1)
      n
       k =
      2
      k=1
    • Properties Summation Notation
      Let S = 1 + 2 + .. + (n – 1) + n, take two copies of the sums and sum them in the following manner:
      S = 1 + 2 + ……+ (n – 1) + n
      S = n + (n – 1) + ……+ 2 + 1
      2S=(n+1)+(n+1)+……+(n+1)+(n+1)
      n times
      n(n + 1)
      Hence 2S = n(n + 1) or S =
      2
      Formula for the Sum of Natural Numbers:
      n(n + 1)
      n
       k =
      2
      k=1
      100
      For example 1+2+3..+100 =  k
      k=1
    • Properties Summation Notation
      Let S = 1 + 2 + .. + (n – 1) + n, take two copies of the sums and sum them in the following manner:
      S = 1 + 2 + ……+ (n – 1) + n
      S = n + (n – 1) + ……+ 2 + 1
      2S=(n+1)+(n+1)+……+(n+1)+(n+1)
      n times
      n(n + 1)
      Hence 2S = n(n + 1) or S =
      2
      Formula for the Sum of Natural Numbers:
      n(n + 1)
      n
       k =
      2
      k=1
      100(100 + 1)
      100
      For example 1+2+3..+100 =  k=
      2
      k=1
      = 5050
    • Properties Summation Notation
      We may use the above properties and the sum formula to sum all linear sums.
    • Properties Summation Notation
      We may use the above properties and the sum formula to sum all linear sums.
      45
      Example E: Find  (2k – 5)
      k=1
    • Properties Summation Notation
      We may use the above properties and the sum formula to sum all linear sums.
      45
      Example E: Find  (2k – 5)
      k=1
      45
       (2k – 5) = Σ2k – Σ5 by property a
      k
      k
      k=1
    • Properties Summation Notation
      We may use the above properties and the sum formula to sum all linear sums.
      45
      Example E: Find  (2k – 5)
      k=1
      45
       (2k – 5) = Σ2k – Σ5 by property a
      k
      k
      k=1
      45
      45
      = 2Σk – Σ5 by property c
      k=1
      k=1
    • Properties Summation Notation
      We may use the above properties and the sum formula to sum all linear sums.
      45
      Example E: Find  (2k – 5)
      k=1
      45
       (2k – 5) = Σ2k – Σ5 by property a
      k
      k
      k=1
      45
      45
      = 2Σk – Σ5 by property c
      k=1
      k=1
      45(45 + 1)
      = 2 – 5*45
      2
    • Properties Summation Notation
      We may use the above properties and the sum formula to sum all linear sums.
      45
      Example E: Find  (2k – 5)
      k=1
      45
       (2k – 5) = Σ2k – Σ5 by property a
      k
      k
      k=1
      45
      45
      = 2Σk – Σ5 by property c
      k=1
      k=1
      45(45 + 1)
      = 2 – 5*45
      2
      by property d
      by the sum formula
    • Properties Summation Notation
      We may use the above properties and the sum formula to sum all linear sums.
      45
      Example E: Find  (2k – 5)
      k=1
      45
       (2k – 5) = Σ2k – Σ5 by property a
      k
      k
      k=1
      45
      45
      = 2Σk – Σ5 by property c
      k=1
      k=1
      45(45 + 1)
      = 2 – 5*45
      2
      by property d
      by the sum formula
      = 2070 – 225 = 1845
    • Properties Summation Notation
      The index variable may be shifted in order to utilize the sum formula.
    • Properties Summation Notation
      The index variable may be shifted in order to utilize the sum formula.
      53
      Example E: Find  (4k – 33)
      k=10
    • Properties Summation Notation
      The index variable may be shifted in order to utilize the sum formula.
      53
      Example F: Find  (4k – 33)
      k=10
      Select a new index, say m, to start at 1.
    • Properties Summation Notation
      The index variable may be shifted in order to utilize the sum formula.
      53
      Example F: Find  (4k – 33)
      k=10
      Select a new index, say m, to start at 1.
      The lower numbers are k = 10 and m = 1  k = m + 9.
    • Properties Summation Notation
      The index variable may be shifted in order to utilize the sum formula.
      53
      Example F: Find  (4k – 33)
      k=10
      Select a new index, say m, to start at 1.
      The lower numbers are k = 10 and m = 1  k = m + 9.
      The upper number is k = 53  53 = m + 9
    • Properties Summation Notation
      The index variable may be shifted in order to utilize the sum formula.
      53
      Example F: Find  (4k – 33)
      k=10
      Select a new index, say m, to start at 1.
      The lower numbers are k = 10 and m = 1  k = m + 9.
      The upper number is k = 53  53 = m + 9 or m = 44.
    • Properties Summation Notation
      The index variable may be shifted in order to utilize the sum formula.
      53
      Example F: Find  (4k – 33)
      k=10
      Select a new index, say m, to start at 1.
      The lower numbers are k = 10 and m = 1  k = m + 9.
      The upper number is k = 53  53 = m + 9 or m = 44.
      Rewrite the sum in terms of m:
    • Properties Summation Notation
      The index variable may be shifted in order to utilize the sum formula.
      53
      Example F: Find  (4k – 33)
      k=10
      Select a new index, say m, to start at 1.
      The lower numbers are k = 10 and m = 1  k = m + 9.
      The upper number is k = 53  53 = m + 9 or m = 44.
      Rewrite the sum in terms of m:
      53
      44
       (4k – 33) = Σ[4(m + 9) – 33]
      m=1
      k=10
    • Properties Summation Notation
      The index variable may be shifted in order to utilize the sum formula.
      53
      Example F: Find  (4k – 33)
      k=10
      Select a new index, say m, to start at 1.
      The lower numbers are k = 10 and m = 1  k = m + 9.
      The upper number is k = 53  53 = m + 9 or m = 44.
      Rewrite the sum in terms of m:
      53
      44
      44
       (4k – 33) = Σ[4(m + 9) – 33] = Σ(4m – 3)
      m=1
      m=1
      k=10
    • Properties Summation Notation
      The index variable may be shifted in order to utilize the sum formula.
      53
      Example F: Find  (4k – 33)
      k=10
      Select a new index, say m, to start at 1.
      The lower numbers are k = 10 and m = 1  k = m + 9.
      The upper number is k = 53  53 = m + 9 or m = 44.
      Rewrite the sum in terms of m:
      53
      44
      44
       (4k – 33) = Σ[4(m + 9) – 33] = Σ(4m – 3)
      m=1
      m=1
      k=10
      44
      44
      = 4Σm – Σ3
      m=1
      m=1
    • Properties Summation Notation
      The index variable may be shifted in order to utilize the sum formula.
      53
      Example F: Find  (4k – 33)
      k=10
      Select a new index, say m, to start at 1.
      The lower numbers are k = 10 and m = 1  k = m + 9.
      The upper number is k = 53  53 = m + 9 or m = 44.
      Rewrite the sum in terms of m:
      53
      44
      44
       (4k – 33) = Σ[4(m + 9) – 33] = Σ(4m – 3)
      m=1
      m=1
      k=10
      44
      44
      = 4Σm – Σ3
      m=1
      m=1
      44(44 + 1)
      = 4 – 3*44
      2
      = 3828