Sequences and Series - PowerPoint by zhangyun

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									  Arithmetic and
Geometric Sequences
and their Summation
14.1 Sequences
Find the next two terms of the following sequences :
(1) 2, 5, 8, 11,…… arithmetic sequence
(2) 2, 6, 18, 54, …. geometric sequence
(3) 2, 4, 8, 16,……. geometric sequence
(4) 5, -25, 125, -625, …. geometric sequence
(5) 3, 4, 6, 9, 13, …….
(6) 5, 2, -1, -4, …..   arithmetic sequence
(7) 0, sin20o, 2sin30o, 3sin40o
 14.1 Sequences
 Consider the following sequence:
 1, 3, 5, 7, 9, ….., 111
1 is the first term of the sequence,mathematically,
                   T(1) = 1 or T1 = 1
3 is the second term of the sequence, mathematically,
                   T(2) = 3 or T2 = 3
5 is the third term of the sequence, mathematically,
                  T(3) = 5 or T3 = 5
111 is the nth term of the sequence, mathematically,
                T(n) = 111 or Tn = 111
14.1 Sequences
Consider the sequence
2, 4, 8, 16, ….
  The sequence is formed from
 timing 2 to the previous term.
So, the sequence can be represented by
the general term
          T(n) = 2n or Tn = 2n
 P.159
Ex. 14A
14.2 Arithmetic Sequence


An arithmetic sequence
(A.S. /A.P.) is a sequence
having a common difference.
14.2 Arithmetic Sequence
Illustrative Examples
14.2 Arithmetic Sequence
14.2 Arithmetic Sequence
14.2 Arithmetic Sequence
 P.166
Ex. 14B
14.2 Arithmetic Sequence
        Arithmetic Means
When a, b and c are three consecutive
terms of arithmetic sequence, the
middle term b is called the arithmetic
mean of a and c.
              ac
           b
               2
14.2 Arithmetic Sequence
      Arithmetic Means
Insert two arithmetic means between 11 and 35.
    T (1)  a  11.......... 1)
                            ....(
    T (2)  a  d .......... 2)
                           .....(
    T (3)  a  2d .......... 3)
                             ...(
    T (4)  a  3d  35.....(4)
14.2 Arithmetic Sequence
Insert two arithmetic means between 11 and 35.

(4)  (1)       3d  24
                 d 8
1st arithmetic mean  a  d  19
2nd arithmetic mean  a  2d  27
 P.170
Ex. 14C
14.3 Geometric Sequence


A geometric sequence
(G.S. / G.P.) is a sequence
having a common ratio.
14.3 Geometric Sequence
Illustrative Examples
14.3 Geometric Sequence
14.3 Geometric Sequence
14.3 Geometric Sequence
 P.176
Ex. 14D
14.3 Geometric Sequence
   Geometric Means
When x, y and z are three
consecutive terms of geometric
sequence, the middle term y is called
the geometric mean of x and z.

          y   xz
14.3 Geometric Sequence
      Geometric Means
Insert two geometric means between 16 and -54.

      T (1)  a  16.......... 1)
                             ....(
      T (2)  aR..........        2
                         .........( )
      T (3)  aR ..........
                   2
                                 3
                          .......( )
      T (4)  aR  54......( )
                   3
                            4
14.3 Geometric Sequence
Insert two geometric means between 16 and -54.
     ( 4)          54  27
            R  3
                      
     (1)           16     8
                 3
            R
                  2
    1st goemetric mean  aR   24
    2nd goemetric mean  aR  36 2
 P.181
Ex. 14E
14.4 Series
 Let’s consider a sequence :
 T(1), T(2), T(3), T(4), …., T(n)
 The expression T(1) + T(2) + T(3) +….+ T(n)
 is called a series. We usually denote the sum
 of the first n term of a series by the notation
 S(n).
  S (n)  T (1)  T (2)  T (3)  .... T (n)
14.5 Arithmetic Series


 Arithmetic Sequence : 2, 5, 8, 11, …

Arithmetic Series : 2 + 5 + 8 + 11 + ….
14.5 Arithmetic Series
   Formula of Arithmetic Series

S(n) = a + a + d + a + 2d +
a + 3d + …. + a + (n - 1)d
                         l
14.5 Arithmetic Series
   Formula of Arithmetic Series

  S(n) = l + l - d + l - 2d +
   l - 3d + …. + a + d + a
 14.5 Arithmetic Series
 S(n) = a + a + d + a + 2d + a + 3d + ………... + a + (n - 1)d


 S(n) = l + l – d + l - 2d + l - 3d + ….+ a + d + a



2S(n) =(a + l)+(a + l)+(a + l)+(a + l)+….. +(a + l)
             2S(n) = n(a + l)
                       n( a  l )
              S ( n) 
                           2
14.5 Arithmetic Series
               n( a  l )
      S ( n) 
                   2
       n
S (n)  [a  a  (n  1)d ]
       2
       n
S (n)  [2a  (n  1)d ]
       2
 P.189
Ex. 14F
14.6 Geometric Series


Geometric Sequence : 3, 9, 27, 81, …

 Geometric Series : 3 + 9 + 27 + 81
14.6 Geometric Series

   Formula of Geometric Series

 S(n) = a + aR +     +  aR2   aR3

        + …. + aRn-1
14.6 Geometric Series
   Formula of Geometric Series


 R.S(n) = aR +     aR
                   +  2   aR 3   +
       aR4 + …. + aRn
 Subtracting two series
 S(n) = a + aR + aR2 + aR3 + …. + aRn-1
R.S(n) = aR + aR2 + aR3 + aR4 + …. + aRn

    S(n) –R.S(n) = a - aRn
     (1 – R) S(n) = a (1 – Rn)

             a(1  R )
                    n
    S ( n)            , where R  1
               1 R
14.6 Geometric Series
           a(1  R )n
  S ( n)            , where R  1
             1 R
Timing –1 on both numerator and denominator

          a( R  1)
                n
 S ( n)            , where R  1
            R 1
 P.196
Ex. 14G
14.6 Geometric Series

Sum to Infinity of a Geometric Series
 14.6 Geometric Series
 Sum to Infinity of a Geometric Series
 Consider such a Geometric Series
       1 1 1 1
              .....
       2 4 8 16
What is the value of common ratio R ?
                   1
                R
                   2
 14.6 Geometric Series
 Sum to Infinity of a Geometric Series
                     1
                  R
                     2
Consider Rn where n tends to the infinity
 1 2          1 3            1 4
( )  0.25   ( )  0.125    ( )  0.0625
 2            2              2

 1 5           1 6            1 7
( )  0.03125 ( )  0.015625 ( )  0.0078125
 2             2              2
                       1 
What will occur for   ( )    if n tends
                       2
to infinity ?
         1 
        ( ) 0
         2
            
        R       0
        where –1< R <1
Summation of a geometric Series to infinity
                
         a(1  R )
S ( )            , where  1  R  1
           1 R

             a
   S ( )       , where  1  R  1
            1 R
 P.203
Ex. 14H
  (extension module)

Summation Notation


       
                             4

Consider the symbol         T (r )
                            r 1


 where T( r ) = 3r + 5

 4

 T (r )
r 1
           = 3(1) + 5 + 3(2) + 5 +3(3)
             + 5 + 3(4) +5
           = 50

								
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