模仿是学会学解题的好方式

20096^ 11! 11–1 ����������� 1 712000 �� ������� !"#$%&�' ()* mn��, 3u.HH 'NÆ���K P'7, .HR*'45u5"s�U�[G1 H 67�5KZ�7���NK%�0,| Q89� #:�e��R*3uKZ"�e� �;<5�~. #KS, mn��Kqr,h= >, 8�0+!S, H.3u�K%�0pJ�. gh 1 F��� {an}�9:a1 = 3 5 , an+1 = 3an 2an + 1 , n = 1, 2, · · · . (1)+{an}�R:?0; (2)Os: �qW� x > 0, an > 1 1 + x − 1 (1 + x)2 ( 2 3n − x ) , n = 1, 2, · · · ; (3)Os: a1 + a2 + · · ·+ an > n2 n + 1 . jhG20086?<6x�8�K�h. | [1]� [2]�jxh�,G73���bl. "K, � (2)lh, �ji� f(x) = 1 1 + x − 1 (1 + x)2 ( 2 3n − x ) , x > 0. �*��1, +"70TGan.[3. �Nqr �((, �Os+�0gh)�Gi�70T� ,+, "dA�qrOBp��, "��K�, �V!� o!5CR!U�3��:Æ. G3Os^ (3)lh, ���N (2)�8�, X.H/� (2)�|R. � (2)��n\℄&T1, 2, · · · , n, Un�[?�+�0, Dq, JU a1 + a2 + · · ·+ an > n 1+x − 1 (1+x)2 ( 2 3 + 2 32 + · · ·+ 2 3n − nx ) , ·································································· (∗) �K (3)a=�WEx, �..�8F�d� (∗)a�x, �m��x�&\G�T, �t�\ GT9? H�qx. y 2 3 + 2 32 + · · · + 2 3n − nx = 0, x = 1 n ( 1− 1 3n ) , 7W (∗)�U a1 + a2 + · · ·+ an > n2 n + 1− 1 3n > n2 n + 1 , UO. ��i�H, ���:�\G��Kq8� �, TUmn8,G7��7℄Ij56q. J >��qr, ��ND, mn.��jxh ()GQj�'�hb. gh2F�i�ft(x) = 1 1 + x − 1 (1 + x)2 (t −x), "�A� tG#�. (1)+i�ft(x)8 (0,+∞)S�70T; (2)�� {an}�, a1 = 3, a2 = 5, "An: 5Sn�[Sn + Sn−2 = 2Sn−1 + 2 n−1 (n > 3), bn = 1 − 1 an . Os: �qW�x > 0, C� bn > f 1 2n (x), n = 1, 2, · · · ; (3)Os: b1 + b2 + · · ·+ bn > n2 n + 1 . ,: (1)�ft(x) = 1 1 + x − 1 (1 + x)2 (t− x), U f ′ t (x)=− 1 (1+x)2 − −(1+x)2−(t−x) · 2(1+x) (1 + x)4 = 2(t − x) (1 + x)3 . VW�x > 0, �., �x < t,, f ′t(x) > 0; �x > t,, f ′t(x) < 0, i1, �x = t,, ft(x) &U70Tft(t) = 1 1 + t . (2)�hW�Sn − Sn−1 = Sn−1 − Sn−2 + 2n−1 (n > 3), an = an−1 + 2 n−1 (n > 3). K H an = (an − an−1) + (an−1 − an−2) + · · · + (a3 − a2) + a2 = 2 n−1 + 2n−2 + · · ·+ 22 + 5 = 1 �� Æ: ������������(08XSYK110) 11–2 20096^ 11! 2n−1 +2n−2 + · · ·+22 +2+1+2 = 2n +1(n > 3). G*KL�, �n = 1� 2,, |m-��, �an = 2 n + 1. �. bn = 1− 1 an = 2n 2n + 1 > 0. y t = 1 2n > 0, ff 1 2n (x) = 1 1 + x − 1 (1 + x)2 ( 1 2n − x ) , � (1)��, f 1 2n (x) 6 f 1 2n ( 1 2n ) = 1 1 + 1 2n = 2n 2n + 1 = bn. �., �qW� x > 0, C�+�0 bn > f 1 2n (x) (n = 1, 2, · · · )��. (3)� (2)�, �qW�x > 0, � b1 + b2 + · · ·+ bn > f 1 2 (x) + f 1 22 (x) + · · ·+ f 1 2n (x), b1 + b2 + · · ·+ bn > n 1 + x − 1 (1 + x)2 ( 1 2 + 1 22 + · · · + 1 2n − nx ) . y 1 2 + 1 22 + · · ·+ 1 2n − nx = 0, �U x = 1 n ( 1 2 + 1 22 + · · · + 1 2n ) = 1 2 ( 1− 1 2n ) n ( 1− 1 2 ) = 1 n ( 1− 1 2n ) . KH b1 + b2 + · · ·+ bn > n 1 + 1 n ( 1− 1 2n ) = n2 n + 1− 1 2n > n2 n + 1 . �I+�0��. gh3 Gx > 0, y > 0. (1)Os: 1 1 + x > 1 1 + y − 1 (1 + y)2 (x−y); (2)Os: C0n 30 30 + 1 +C1n 31 31 + 1 +C2n 32 32 + 1 + · · ·+ Cnn 3n 3n + 1 > 3n · 2n 3n + 2n . jhH20096?<�KM�N�7h, ig h1aan = 3n 3n + 2 , ./e/+�0�|�, 3 :�8v�����h�. KH, mn7��. �Y'qx, �+��gh1�, ��^O� K,G�gh9? R*���7,� oet7 P, .%�3QjOs��. Os: (1)I+�0��K 1 1 + x > 1 1 + y − 1 (1 + y)2 [(1 + x) − (1 + y)] (q1e1HG3(>\E�|�) = 1 1 + y − 1 + x (1 + y)2 + 1 1 + y (\Oq88�aU�3 ��UÆ) = 2 1 + y − 1 + x (1 + y)2 , -.H 1 1 + x > 2 1 + y − 1 + x (1 + y)2 , ��K1 > 2 ( 1 + x 1 + y ) − ( 1 + x 1 + y )2 , (G301 (%) ��K ( 1 + x 1 + y − 1 )2 > 0, UO. (2)8+�0 (1)a, &x = 1 3k , (� (1)5 (2) %&�R:1s2&\T���) U 3k 3k + 1 > 1 1 + y − 1 (1 + y)2 ( 1 3k − y ) , v&[�.Ckn, (�℄�℄ QR, .9S LT�, rPbMAH) U Ck n 3k 3k + 1 > 1 1 + y Ck n − 1 (1 + y)2 ( 1 3k Ck n − yCk n ) , y&k = 0, 1, 2, · · · , n, Un + 1�+�0, v&Dq, U n∑ k=0 Ck n 3k 3k + 1 > 1 1 + y n∑ k=0 Ck n − 1 (1 + y)2 ( n∑ k=0 Ckn 1 3k − y n∑ k=0 Ckn ) , VW� n∑ k=0 Ckn = 2 n, n∑ k=0 Ckn 1 3k = ( 1 + 1 3 )n = ( 4 3 )n , i1, �+�0 n∑ k=0 Ckn 3k 3k + 1 > 2n 1 + y − 1 (1 + y)2 [( 4 3 )n − 2ny ] , KH, y ( 4 3 )n − 2ny = 0, y = 2n 3n , 7W S$�+�0, �U C0n 30 30 + 1 + C1n 31 31 + 1 + C2n 32 32 + 1 + · · ·+ Cnn 3n 3n + 1 > 3n · 2n 3n + 2n . ,h\4,, ��℄$��hbI�GFG , *�gh, R*�htfh�“�f”, 3u fh�,G��, .^O' ���gh. 3u 20096^ 11! 11–3 ��'4� �8,��iUV��V.ivo,3 u� �K-,3u, y�Wj,3u�XY, � ���iÆ��N-\Z<�()�H+. , h\4,,���[{bM,KZ�8,x��� �?, 0\�℄��. /℄�H�&2�(d )GmnCU�,hK^. _301+�Hb 30, `>q!, +�<�t℄a\4ght, gh�7�, b)qrst. 1234 [1] d�. ��6x�K�h�,�, l [J]. �KRe, 2008, (9): [2] d�, faM. ��7�+�0�eh , [J]. �KRe, 2008,(10): [3]gth. �K,hK%m [M]. < : ? <-�0K>{i, 1997.