1 7 7

: ... Sequences and Series

  1. #1

    Jul 2011
    -
    4,543

    ... Sequences and Series




    ( ) SEQUENCES

    : . . . ( ) .
    : : 2 4 6 8 . . . .
    2 4 . . . .
    : : 1 3 5 . . . = 1 = 3 . . . .


    : GENERAL TERM OF A SEQUENCE
    . n un.
    : n2 + 3 = un
    : u1, u2, u3 . . . n: 1 2 3 . . .

    5 = 2 1 + 3 = u1
    7= 2 2 + 3 = u2
    9= 2 3 + 3 = u3
    : 5 7 9 . . .

    :

    :
    :
    )
    ) 1 8 27 . . . .
    )7 7 7 . . . . .


    SERIES AND SIGMA NOTATION
    (()) (()) ((+)) : 2 5 8 . . . : 2 + 5 + 8 + . . . ∑ ( )

    : = a = b a b .
    :
    30= 3 1 + 3 2 + 3 3 + 3 4 = 3 + 6 + 9 + 12 = a
    10= 1 + 2 + 3 + 4 = b
    B3 = a
    :

    1) z n = z∑ n n z

    2) (∑u)a =au∑ a

    3) (u + d) ∑= ∑ u + ∑d



    :ARITHMETIC PROGRESSION

    6 50 75 100 . . .
    / .
    25 :50 75 100 125 150 175.
    175 :
    50 + 75 + 100 + 125 + 150 + 175 = 675 .
    : 50 75 100 125 150 175 .
    :

    .


    :
    :
    1) 3 5 7 . . . 31 2) 1 + 1/2 + 1/3 + . . . + 1 /20
    3) 4) r + 7 ) ∑
    r=1
    :
    1) 3 5 7 . . . 31 5 3 = 2 7 5 = 2
    2) 1 + 1/2 + 1/3 + . . . + 1 /20
    1 / 2 1 = - 1 / 2 1/3 1/2 = - 1/6
    3) : 2 3 5 7 . . . 3 2 = 1
    5 3= 2
    4) (r + 7 ) ∑ = (1+7) + (2+7 ) + (3+7 ) + ( 4 + 7) + . . . + ( 10+ 7)
    = 8 +9+10+. . . +17
    9 - 8= 10 - 9 = 1


    :
    : 3 7 11 15 19 23 27 4 :
    7 = 3 + 41 = 3 + ( 2 1 )4 = u2
    11 = 3 + 4 2 = 3 (3 1 ) 4 = u3


    27 = 3 + 6 4 = 3 + (7 1 ) 4 = u7
    = 3 ( ) + ( 1 ) .
    : a d = a+ d = a + d2 = a+ 9d . . . ( )
    d d ( n 1 ) + a = un


    ( n 1 ) d = a = un
    :
    2 5. .
    :
    2= a 5 = d
    a+ ( n 1 ) n= un

    2 + (5 1 ) 5=u5
    22= 2 + 4 5 = u
    : 2 7 12 17 22
    = 22 .


  2. #2

    Jul 2011
    -
    4,543

    : ... Sequences and Series

    :ARITHMETIC MEAN
    :
    a b = a + b / 2
    a a + b /2 b . a b : x1 x2 x 2 . . . xn a b .








    : 5 9 13 17 9 5 13 9 = 5 = 13 / 2 13 9 17.

    :
    4 4 29
    : 4 4 29 : 4 x 1 x2 x3 x4 29
    4= a 29 = u6
    a + d 5 = u6
    29= 4 + 5 d
    20= 5d d = 5
    : 4 9 14 19 24 29 9 14 19 24
    :SUM OF ARITHMETIC SERIES
    1787 1 100
    1+ 2 + 3 + . . . + 100
    ( ) 5050.
    :
    cn = 1 + 2 + 3 + . . . + 100
    cn = 100 + 99 + 98 + . . . + 1
    cn 2= 1010 + 101+ 101 + . . . + 101 ( 100)
    2c = 101 100
    :
    n/ 2 ( a+ k) = cn . . . (1) n / 2 (2a + (n 1 )d ) = cn . . . (2)



    c= 101 100 / 2 = 50 ( 101 ) = 5050


    : 20 : 3 + 8 + 13 + . . .
    : a= 3 d = 5
    n / 2 (2a + ( n 1 ) d ) = cn
    20 / 2 ( 2 3 + 19 5 ) = 10 ( 6 + 95 ) = cn
    = 10 101 = 1010






    GEOMETRIC SEQUENCE AND SERIES

    :
    .
    .













    :
    a r
    : r 1 - aN = un


    :
    1215 : 5 15 45 . . .
    :
    1215 un a = 5 r = 3
    1215 = 1-arn = un
    1215 = 5 3n-1 ( 5 )
    234 = 3n- 1 n 1 = 5
    n = 6
    1215
    :GEOMETRIC MEAN
    :
    a b c a c b .

    : 4 9
    : 4 c 9
    4 c 9 c / 4 = 9 / c
    c2 = 36 = c= 6
    4 9 6 : 4 6 9 : 4 -6 9
    6 = 4 9
    :
    c a b c2 = ab





    :SUM OF GEOMETRIC SERIES

    n a r :

    r* 1








    1 = r :

    a+ a + a + . . . = a ∑ = na

    ( 1 ) :
    : 3 + 6 + 12 + . . .
    :
    3= a 2 = r

    c = 3 ( 2 6 1 ) / 2 1 = 3 ( 64 1 ) / 1 = 3 63 = 189
    ( 2 ) 4
    5R
    R=1
    :
    : 5 1 + 5 2+ 5 3+ 5 4
    : 5 + 25 + 125 + 625 = 780
    5 5 :
    c = 5 ( 5 4 1 ) / 5 1 = 5 624 / 4 = 5 156 = 780
    :

    a r : a/ 1 r r<1.


  3. #3

    Jul 2011
    -
    4,543

    : ... Sequences and Series

    :
    INFINITE GEOMETRIC SERIES


    :
    1 + 1/3 +1/9 + . . . ( ).
    :
    = 1 = 1/3
    1/3< 1
    c= a / 1 r = 1 / 1-1/3 = 1 / 2/3 = 3\2
    :
    ( -1/2)r ∑
    :
    -1/2 = (-1/2) =u
    u2= ( -1/2)2 = 1/4 u3= (-1/2)3=-1/8
    : -1/2 1/4 -1/8 . . . .
    r = 1/4 -1/2 = - 1/2 | -1/2| = 1/2<1
    :
    c = a/1- r = -1/2 /( 1+1/2 ) = -1/2 / 3/2 = -1/3
    :
    (3):
    4,
    : 00000 444, 0
    00000 444, 0 = 4/10 + 4/100 + 4/ 1000 +. . .
    4/100 4/10 = 1/10 4 / 1000 4/ 100 = 1/10
    r = 1/10 4/10 =a
    c= a / 1- r = 4/ 10 / 1-1/10 = 4/9 9/10 =
    4/10 10/9 = 4/9
    4, = 4/9 . 4/9

    ( ) SEQUENCES : . . . ( ) . : : 2 4 6 8 . . . . 2 4 . . . .: : 1 3 5 . . . = 1 = 3 . . . . : GENERAL TERM OF A SEQUENCE . n un.: n2 + 3 = un : u1, u2, u3 . . . n: 1 2 3 . . . 5 = 2 1 + 3 = u1 7= 2 2 + 3 = u2 9= 2 3 + 3 = u3 : 5 7 9 . . .


  4. #4

    Jul 2011
    -
    4,543

    : ... Sequences and Series

    :
    : :) 1 8 27 . . . .)7 7 7 . . . . . SERIES AND SIGMA NOTATION (()) (()) ((+)) : 2 5 8 . . . : 2 + 5 + 8 + . . . ∑ ( ) : = a = a b . :30= 3 1 + 3 2 + 3 3 + 3 4 = 3 + 6 + 9 + 12 = a10= 1 + 2 + 3 + 4 = b B3 =a
    :ARITHMETIC PROGRESSION 6 50 75 100 . . . / . 25 :50 75 100 125 150 175. 175 :50 + 75 + 100 + 125 + 150 + 175 = 675 . : 50 75 100 125 150 175 . : :1) 3 5 7 . . . 31 2) 1 + 1/2 + 1/3 + . . . + 1 /203) 4) r + 7 ) ∑ r=1 :1) 3 5 7 . . . 31 5 3 = 2 7 5 = 22) 1 + 1/2 + 1/3 + . . . + 1 /20 1 / 2 1 = - 1 / 2 1/3 1/2 = - 1/63) : 2 3 5 7 . . . 3 2 = 1 5 3= 2 4) (r + 7 ) ∑ = (1+7) + (2+7 ) + (3+7 ) + ( 4 + 7) + . . . + ( 10+ 7) = 8 +9+10+. . . +17 9 - 8= 10 - 9 = 1






  5. #5
    Guest   abutaani
    Sep 2013
    48

    : ... Sequences and Series


  6. #6

    Sep 2014
    9


  7. #7

    Mar 2015
    39


: 1 (0 1 )

  1. : 10
    : 02-01-2017, 02:02 PM
  2. : 41
    : 31-12-2016, 01:46 PM
  3. : 73
    : 07-12-2016, 07:38 PM
  4. : 10
    : 28-11-2016, 09:32 PM
  5. : 1
    : 08-11-2011, 10:02 AM

: 0