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					                                                                                                 ‫( ا‬un ) n≥1                            ‫ا‬               1
                      2u n − 1
               vn =                      ‫ و‬u n +1 = 2u n + 3                      ‫و‬        u0 = 0
                      2u n + 6                          2u n + 7
                                                                                                                         1
                           (u n )         ‫(ادرس ر‬b                                         . (∀n ∈ IN ); − 3 < u n <       ‫أن‬                           (a (1
                                                                                                                         2
                                                                                   . (∀n ∈ IN ):u n +1 − 1 ≤ 1 (u n − 1 ) ‫أن‬                             (a (2
                                                                                                                   2     8              2
                                                                       . lim u n                 ‫وا‬        ‫ر‬             (u n ) ‫أن‬                     ‫( ا‬b
                                     .‫و ه ا ول‬                ‫دأ‬              ‫( ه‬v n )                  ‫أن ا‬                                              (a (3
                                        . limv n ‫ و‬lim u n         ‫ وا‬n          (u n )          (v n )                                                 ‫( ا‬b
                                                        . lim S n ‫ و‬S n = v0 + v1 + ............. + v n                                                ‫( أ‬c
                                                                                                 Pn = v 0 .v1 ...............v n                       ‫( أ‬d
             1                       u 0 = 3
    vn =                         ‫و‬                        :                                ‫( ا‬un ) n∈IN                            ‫ا‬                   2
           un − 2                    u = 5u n − 4
                                      n +1
                                             un + 1
                                                                                                           . (∀n ∈ IN ); un > 2 ‫أن‬      (1
                                                                                                              (u n )       ‫ا‬      ‫2( ادرس ر‬
                                                                                . lim u n                   ‫ر وا‬      (u n ) ‫أن‬        ‫3( ا‬
                                       .‫ه ا ول‬             ‫و‬                      ‫دأ‬                           (v n )        ‫أن ا‬    (a (4
                                                                       .‫ى‬         ‫أ‬                  (u n )                    ‫ا‬                            ‫( أ‬b
                                                                6x
                                               f ( x) =             :                                               ‫ا‬f        ‫ا ا‬                       3
                                                               x +43

                                                                                                      .f ‫ا ا‬                                            ‫د‬      (1
                                                       .               ‫ﺡ‬               ‫ل‬                   [
                                                                                                     ‫2 3 ,0 ﺡ‬            ]                       f ‫أن‬          (2
                                                       u 0 = 1
                                                                         : ‫ـ‬                                      ‫( ا‬u n ) n∈IN                   ‫ا‬               (3
                                                       u n +1 = f (u n )
                                                                                       . (∀n ∈ IN ) : 1 ≤ u n < 3 2 : ‫أن‬                                      (a
                                          . lim un                         ‫وا‬      ‫ر‬             ‫أ‬                  ‫وا‬          ‫ا‬           (un ) ‫أن‬           (b
                                              3
u n +1 = u n − 3u n + 4
           2
                             ‫و‬         u0 =             :                                   ‫( ا‬un ) n∈IN                            ‫ا‬                   4
                                              2
                                                       f ( x) = x 2 − 3 x + 4                        :                                      ‫ا‬f   ‫ا ا‬
                                                                                            . f ([1,2]) ⊂ [1,2] ‫أن‬                                        (a
                                                                                   . (∀x ∈ IR) : f ( x) ≥ x ‫أن‬                                            (b
                                                  .        ‫ﺹ‬            (u n ) ‫∀( وأن‬n ∈ IN ) : 1 ≤ u n ≤ 2 ‫أن‬                                          ‫( ا‬c
                                                                       . lim u n                     ‫وا‬        ‫ر‬             (u n ) ‫أن‬                  ‫( ا‬d
                                  u0 = 4
                                 
                                         2u 2 − 3             :                                 ‫( ا‬u n ) n∈IN                              ‫ا‬           5
                                 u n +1 = n
                                         un + 2
                                                                                             .       (∀n ∈ IN ) : u n > 3 : ‫أن‬                                 (1

                                                                                                         . (u n )                       ‫ا‬        ‫2( أدرس ر‬
                                                                                                   3
                                                                        . (∀n ∈ IN ) : u n +1 − 3 > ( u n − 3) : ‫أن‬                                            (3
                                                                                                   2
                                                                                                        3
                                                                                    (∀n ∈ IN ) : u n ≥ ( ) n + 3 ‫أن‬                                               ‫4( ا‬
                                                                                                        2
                                                                                            ‫ر ؟‬         (u n )                                           ‫ا‬     ‫5( ه‬
                      1 3
       u n +1 =   3     un + 2         ‫و‬       u0 = 1 :                                      ‫( ا‬u n ) n∈IN                            ‫ا‬                       6
                      3
                                                             (u n )         ‫∀( 2( ادرس ر‬n ∈ IN ) : u n ≥ 1 ‫: أن‬                                                     (1
                                                                           . lim u n    ‫ر وا‬       (u n ) ‫أن‬                                                      ‫3( ا‬
                                                        vn = u 3 − 3                                     ‫( ا‬v n ) n∈IN                               ‫ا‬               (4
                                                              n

                                                                                                      ‫( ه‬v n )                      ‫أن ا‬                        (a
                                                                                           n           (u n )                 (v n )                          ‫( ا‬b
             (3n + 3)u n − 8n − 12
  u n +1 =                         ‫و‬            u1 = 1 :                                       ‫( ا‬u n ) n∈IN *                            ‫ا‬                   7
                       n
                                                       (u n )         ‫∀( . 2( ادرس ر‬n ∈ IN * − {1}) : u n ≤ 0 ‫أن‬                                                     (1
                                                                      u
                                                 .            vn = 4 − n                                 ‫( ا‬v n ) n∈IN                               ‫ا‬               (3
                                                                     n
                                       .‫ه ا ول‬           ‫و‬                     ‫دأ‬                     ‫( ه‬v n )                                ‫أن ا‬             (a
                                             . lim v n ‫ و‬lim u n                      ‫ وا‬n                 (u n )             (v n )                          ‫( ا‬b

                                                     1
u n +1 = 1 − 3 5 − 3u n            ‫و‬       u0 = −        :                                  ‫( ا‬un ) n∈IN                          ‫ا‬                           8
                                                     3
                                                       (u n ) ‫∀( . 2( ادرس ر‬n ∈ IN ) : − 1 < u n < 0 ‫أن‬       (1
                                                                                      3 2
                                                        . (∀n ∈ IN ) : 0 < u n + 1 < ( ) n .     : ‫أن‬   ‫( ا‬a (2
                                                                                      4 3
                                                            . lim u n          ‫ر وا‬          (u n ) ‫أن‬   ‫( ا‬b
                          un + 2
                           3
               u n +1 =            ‫و‬       u0 = 1            :                              ‫( ا‬un ) n∈IN                          ‫ا‬                           9
                          un + 1
                           2


                                                              . (u n )       ‫∀( . 2( ادرس ر‬n ∈ IN );0 <u n < 2 ‫أن‬                                                   (1
                                                                           . lim u n      ‫ر وا‬       (u n ) ‫أن‬                                                    ‫3( ا‬
                                                                                 . (∀n ∈ IN ): 2 − u n +1 ≤ 4 (2 − u n ) ‫أن‬                                   (a (4
                                                                                                                     5
                                             lim u n                  ‫وا‬        ‫ر‬        (u n ) ‫ى أن‬             ‫أ‬                                       ‫( ا‬b
                                                                                               S n = v0 + v1 + ............. + v n                                   (5
                                                                                                              4
                                                                                (∀n ∈ IN ): S n ≥ 2n − 3 + 5( ) n +1 : ‫أن‬                                     (a
                                                                                                              5
                                                                                                                         lim S n                             ‫( وا‬b
                      1   2                u0 = 3
      u n +1 = 2 +      − 2        ‫و‬                      :                                ‫( ا‬un ) n∈IN                       ‫ا‬                          10
                      un un

                                                 (u n )                    ‫ا‬        ‫∀( . 2( ادرس ر‬n ∈ IN ) ; un > 2 ‫أن‬                                               (1
                                                                               . lim u n              ‫وا‬      ‫ر‬            (u n ) ‫أن‬                              ‫3( ا‬
                                                                                     . (∀n ∈ IN ): un +1 − 2 ≤ 1 (un − 2) ‫أن‬                                         (4
                                                                                                                          4
                                                . lim u n                      ‫وا‬    ‫ر‬         (u n ) ‫ى أن‬                ‫أ‬                                       ‫5( ا‬
                                             f ( x) = ( x + 1 − 1) 3 : ‫ـ‬                    ‫ا‬f ‫ا ا‬          11
                                                                                         . f ‫ا ا‬          ‫د‬      (1
‫د‬   (b         .    ‫ﺡ‬               J‫ل‬               ‫ل [∞+,1 −[ ﺡ‬                      ‫ا‬       f ‫أن ا ا‬      (a (2
                                                                                                 .J     x      f −1 ( x)
                                                       3             :                        ‫( ا‬u n ) n∈IN           ‫ا‬             (3
                                              u0 = −
                                                       4
                                               un +1 = f (un )
                                              
                                                                                   (∀n ∈ IN ) : − 1 < u n < 0 ‫: أن‬                (a
                                                                                   .   ‫( ا‬u n )            ‫أن ا‬                   (b
                                         (   t = x +1 −1                  ) f ( x) = x  ‫د‬     ‫[∞+,1 −[ ا‬                          (c
                                                                           lim u n      ‫ر وا‬        (u n ) ‫أن‬                     (d
                                                      1
                               . f ( x) =                  :                                      ‫ا‬f      ‫ا ا‬               12
                                                    4x + 4
                                                         2

                                           1
                                   u 0 =                    :                           ‫( ا‬u n ) n∈IN             ‫ا‬        ‫و‬
                                           2
                                    u n +1 = f (u n )
                                   
                                                                                     . IR +       f ‫ات‬               ‫( ادرس‬a (1
                                     .            1
                                             α ∈  0,           ‫ا‬        ‫و‬             f ( x) = x ‫د‬               ‫أن ا‬    (b
                                                  2

                                                    . (∀( x, y ) ∈ [0,1]2 ) : f ( x) − f ( y ) ≤ 1 x − y ‫أن‬                     (c
                                                                                                           2
                                                                                      . (∀n ∈ IN ) : 0 < u n ≤ 1       ‫أن‬            (2
                                                                                                                   2
                                                                                                        1
                                                                 (∀n ∈ IN ) : u n +1 − α ≤                u n − α ‫أن‬         (a (3
                                                                                                        2
                               .                                     ‫وا‬       ‫ر‬    (u n ) n∈IN                ‫أن ا‬         ‫( ا‬b
                              u0 = 0
                                       :                                           ‫ ( ا‬un )               ‫ا‬                13
                        un +1 = 6 − un
                                                         . f ([0,6]) ‫د‬               ‫ و‬f ( x) = 6 − x ‫ات‬                   ‫1( ادرس‬
                                                          . (∀n ∈ IN ) : 0 ≤ un ≤ 6 ‫أن‬              *
                                                                                                                                     (2
                                                                 wn = u2 n +1 ‫ و‬vn = u2 n                                            (3
                    .      ‫ﺹ‬         wn ‫و‬    ‫( ا‬vn ) ‫∀( وأن‬n ∈ IN ) vn ≤ wn ‫أن‬
                                                                             1
    . lim un       ‫وا‬     ‫ر‬         (un ) ‫أن‬     ‫∀( وا‬n ∈ IN ): un +1 − 2 ≤ un − 2 ‫أن‬                                                (4
                                                                             2
                                    . ‫آ‬       ‫ا‬        ‫ ( ﺡ د ن و د‬wn ) ‫( و‬vn ) ‫أن‬                                                   (5


                         u0 ≥ 3 a
                  
           a > 0 :        1       a :                                                ‫ ( ا‬un )                 ‫ا‬            14
                   un +1 = (2un + 2 )
                          3       un
                  

                                                                                      . ( ∀n ∈ IN ): un > 0 ‫أن‬                  (a (1
                                                                                   ( 2 un + 3 a )
                                     ( ∀n ∈ IN ): un +1 − 3 a =                            2
                                                                                                  (un − 3 a ) 2 ‫أن‬              (b
                                                                                        3un
                                                                 .        ‫ر‬       (un ) ‫أن‬   ‫ و‬un ‫ 3 و‬a      ‫رن‬                 (c
                                                                                            2
                                         . lim un                         ‫ . وا‬un +1 − 3 a − (un − 3 a ) ≤ 0 ‫أن‬                      (2
                                                                                            3
               1
(∀n ∈ IN ) :     < xn < 1                        ‫وﺡ‬          ‫ر‬                         ( xn ) n∈IN                                                 15
               2
                                    u 0 = x0
                                                                                             ‫( ا‬u n ) n∈IN
                                   u = u n + x n +1 :                                                                                     ‫ا‬         ‫و‬
                                    n +1 1 + u n x n +1
                                   
                                                             . 1 ‫( ﺡ ودة ـ 0 و‬u n )                                                 ‫أن ا‬            (a (1
                                      . ‫ر‬        (u n ) ‫أن‬                    ‫وا‬           ‫ا‬      (u n )                             ‫أن ا‬           (b
                                                     (∀n ∈ IN ) : x n (−1 + u n u n −1 ) = u n −1 − u n ‫أن‬                                          (a (2
                                                                                                             lim u n                               ‫( ا‬b
                          :                          ‫( ا‬v n ) n∈IN ‫و‬               (u n ) n∈IN                          ‫ا‬                          16
                                    u 0 = a , v0 = b
                                   
                 (a < b          )          u n + 2v n             u + 3v n
                                   u n +1 =
                                                3
                                                        , v n +1 = n
                                                                        4
                                                                . (∀n ∈ IN ) : 0 < u n < v n : ‫أن‬                                                   (a (1
                                                   . ‫ﺹ‬         (v n ) ‫و‬     ‫( ا‬u n )            ‫أن ا‬                                                 (b
                                                                 .‫ر ن‬        (v n ) ‫( و‬u n ) ‫أن‬                                                    ‫( ا‬c
                                                                            t n = 3u n + 8v n ‫ . و‬wn = v n − u n                                           (2
                                                                      (t n ) ‫و‬                   ‫ ( ه‬wn )                             ‫أن ا‬           (a
                                            lim u n‫ و‬lim v n                   ‫أ‬        .n                  vn ‫ و‬u n                               ‫( ا‬b
                                  :                      ‫ ( ا‬wn) ‫( و‬vn) (u n )                        ‫ت‬                         ‫ا‬                  17
                                                          1          u 0 = 20 ; u1 = 6
                                             vn = un +1 + 4 un      
                                                                               1        1
                                                           1
                                            wn = un +1 − un         u n +1 = − 20 u n + 20 u n −1
                                                                     
                                                          5
                                                            .          ‫ه‬              (wn)‫( و‬vn) ‫أن‬                                                     (1
                                              . limun           ‫ . وا‬n         un      vn ; wn                                                       ‫2( ا‬
                                      . limSn          ‫ وا‬un          Sn = u0 + u1 +........+ un                                                      ‫3( ا‬
           1
u n +1 =     (u n + u n + 2 ) ‫و‬         u0 = 1 :                                    ‫( ا‬un ) n∈IN                            ‫ا‬                      18
           2
                                                                                            . (∀n ∈ IN );1 ≤u n < 4 ‫أن‬     (1
                                                                                               . (u n )       ‫ا‬     ‫2( ادرس ر‬
                                                                     . (∀n ∈ IN ): 0 < 4 − u n +1 ≤ 3 (4 − u n ) ‫أن‬                                 (a (3
                                                                                                             4
                                                         . limun                         ‫وا‬       ‫ر‬         (un) ‫أن‬                                 ‫( ا‬b
                                          5
                                     u0 =
                              
                                           2     :                                  ‫ ( ا‬un ) n ≥ 0                  ‫ا‬                              19
                                       1
                              un +1 = (un + n )
                                              2

                                      3
                                              n 2 − 3n + 3
               ( ∀n ∈ IN ) : vn = un − (                         )      :                                 ‫( ا‬vn )                              ‫ا‬                ‫و‬
                                                     2
                                       . ‫ه ا ول‬          ‫و‬                  ‫دأ‬                        ‫ه‬                             (vn ) ‫أن‬               (a
                                                                                           . n                   un                   vn                 ‫( ا‬b
 u0 = 2

           un           :                             ‫ ( ا‬un )                         ‫ا‬                                          20
u n +1 = 3 + 2u
                n

                                                                .       ‫ﺹ‬               ‫و‬                    (un ) ‫أن‬                       (1
                                                  un + 1
                         (∀n ∈ IN ):vn =                       :‫ب‬                       ‫( ا‬vn )                                ‫ا‬            (2
                                                   un
          . ‫ه ا ول‬               ‫و‬               ‫دأ‬        ،                    ‫ه‬                        (vn ) ‫أن‬                    (a
                                     . lim un                  ‫ وا‬n                         un               vn                    ‫( ا‬b
                                                                                    n
                                                                                        1
                                                                 . Sn = ∑ u                      n                                         ‫3( ا‬
                                                                                k =0        k


       u0 = 1
      
               1 2                     :‫ب‬                 ‫( ا‬u n ) n∈IN                             ‫ا‬                             21
      u n +1 = 2 u n + 12
      
                                   v n = un − 4 : ‫ب‬
                                          2
                                                             ‫( ا‬vn )                                                   ‫ا‬               ‫و‬
          . ‫ه ا ول‬               ‫و‬        ‫و دأ‬             ‫( ه‬v n )                                                ‫أن ا‬                     (1
                                    . lim un         ‫ وا‬n           un                                            vn                       ‫2( ا‬
                                        .n        S n = v 0 + v1 +                                           + vn                          ‫3( أ‬
      u0 = 2

u = u n + u n
           2
                             :                              ‫( ا‬u n ) n∈IN                                ‫ا‬                         22
 n +1    un + 1
             2

                                                        . (∀n ∈ IN ) : 1 < u n ≤ 2 ‫أن‬                                                 (1
                                                      . ‫ﺹ‬      (u n )          ‫أن ا‬                                                (a (2
                                      . lim u n                 ‫ر‬   ‫أ‬  (u n ) ‫أن‬                                                   (b
                                                                1
                                     . (∀n ∈ IN ) : u n +1 − 1 ≤ (u n − 1) : ‫أن‬                                                    (a (3
                                                                2
            lim u n                   ‫أ‬     ‫ر‬       (u n ) ‫أ ى أن‬                                                                  ‫( ا‬b
                         x2 + 4
              f ( x) =          :                                           ‫ا‬f          ‫ا ا‬                                        23
                          2x
                                                                                    f ([2,3]) ⊂ [2,3] ‫أن‬                                    (1
                       u1 = 3
                      
                      u = u n + 4                    :‫ـ‬                    ‫( ا‬u n ) n∈IN *                                ‫ا‬                (2
                               2

                       n +1   2u n
                      
                                           .2‫د‬             ‫( ﺹ رة‬u n )                                        ‫أن ا‬                     (a
                                                          . ‫ﺹ‬   (u n )                                        ‫أن ا‬                     (b
                                           . lim u n                    ‫أ‬               ‫ر‬                (u n ) ‫أن‬                         (c
                          6x
              f ( x) =        :                                             ‫ا‬f          ‫ا ا‬                                        24
                         x +4    3

                                                                     .f ‫ا ا‬                                                        ‫د‬        (1
                             .         ‫ﺡ‬              ‫ل‬             ‫2 3 ,0 ﺡ‬[               ]                              f ‫أن‬             (2
                         u 0 = 1
                                           : ‫ـ‬                                 ‫( ا‬u n ) n∈IN                                  ‫ا‬            (3
                         u n +1 = f (u n )
                                                  . (∀n ∈ IN ) : 1 ≤ u n < 3 2 : ‫أن‬                                                 (a
          . lim un                    ‫وا‬     ‫ر‬        ‫أ‬        ‫وا‬       ‫( ا‬un ) ‫أن‬                                                  (b
                              u0 = 0   ; u1 = 1
.   v n = u n +1 − u n   ‫ و‬
                                    3       1                    ‫( ا‬u n ) n≥0           ‫ا‬             25
                          un + 2   = un +1 − un
                                    2       2

                                        . ‫ه ا ول‬   ‫و‬        ‫دأ‬   ‫و‬           ‫( ه‬v n )               ‫أن ا‬      (1
                                                                  .n          un              vn              (2
                                                                                        . lim u n           ‫3( أ‬
                                                       .n        S n = u 0 + u1 +            + un           ‫4( أ‬

				
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