51 bài toán 7 vmo 1990 4 cho a 2 b 1 l p hai dãy s

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Bài toán 7. ( VMO-1990 [4]) Cho a 0 = 2 , b 0 = 1 . L p hai dãy s ( a n ) , ( b n ) v i n = 0, 1, 2, ... theo quy t c sau: 2 a n . b n a n = ; b = a . b + 1 a n + b n n + 1 n + 1 n Ch ng minh r ng các dãy ( a n ) , ( b n ) có cùng m t gi i h n khi n . Tìm gi i h n đó. Ta chú ý: a 0 = 2 = 1 1 , b 0 = 1 L i gi i. a 1 = 2 a 0 b 0 1 cos 2 3 2 2 = = = 1 ; b 1 = a 1 b 0 = 1 a 0 + b 0 1 1 a 0 + b 0 cos + 1 3 cos 2 6 cos 6 T đó, b ng quy n p, ta ch ng minh r ng: a n = cos 2 . 3 b n = cos 2 . 3 . cos 2 2 . 3 . cos 2 2 . 3 ... cos 2 n 1 . 3 ... cos 2 n 1 . 3 . cos 2 n . 3 . cos 2 n . 3 Σ 1 Σ−1 n 1 L u ý r ng: cos ư . 2 . 3 cos ... 2 2 . 3 cos . 2 n 1 . 3 cos = 2 n . 3 2 n sin 3 n 1 2 n . sin 2 n . sin . sin 2 n . 3 Ta có: a n = sin 2 n . 3 3 . cos 2 n . 3 ( 1 ) ; b n = 2 n . 3 ( 2 ) sin 3 T (1), (2) t n t i lim a n và lim b n Ngoài ra: n lim a n = lim . . n
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2 n . sin 2 n . 3 3 = = n n sin 9 sin 3 . cos 2 n . 3 lim b n = lim a n . lim cos 3 2 3 = n n n 2 . 3 9 2 3 V y hai dãy ( a n ) , ( b n ) có cùng gi i h n chung là 9 Bài t p đ ngh : 2 3 n
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Bài 1: Cho hai dãy ( a n ), ( b n ) nh sau: a < b ư cho tr c: ướ a 1 = a + b ; b 1 = a . a 1 2 a 2 = a 1 + b 1 ; b 2 = a 2 . b 1 ... a n = a n 1 + b n 1 ; b n = a n . b n 1 2 a. Tìm lim b n , b.Tìm lim a n n n H ư ng d n: Đ t cos = a , . 0 < < Σ b B ng quy n p ta d dàng có: a = b . cos ... cos . cos 2 b . sin cos . = 2 n b n = b . cos 2 cos 2 n 1 . cos 2 2 n = b . sin . 2 n . sin 2 n b sin b.Ta cũng có: lim b n a n = b n . cos 2 n lim a n = b sin . lim cos b sin = n n 2 n Bài 2: Tìm lim . 2 2 . . 2 2 + 2 ........... . 2 . 2 + 2 + ... + 2 Σ (th a s c i có n d u căn) ướ Bài 3: Cho dãy s : x 1 = t x n + 1 = 4 x n ( 1 x n ) Tìm các giá tr c a t đ x 1998 = 0 ? H ng d n: ướ Phân tích 4 x n ( 1 x n ) = 1 − ( 2 x n 1 ) 2 đ ch ng minh | x n | ≤ 1 2 2 n 2 n 1 2 2 sin 2 n 2 . . ... = .
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Đ t t = sin 2 a suy ra công th c xác đ nh x n Bài 4: Cho dãy s : x 1 = t x n + 1 = 2 x 2 1 . n
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Tìm công th c t ng quát xác đ nh x n . H ng d n: ướ Ta xét hai tr ng h p: ườ | a | ≤ 1 và | a | ≥ 1 2.5. L p các bài toán v gi i h n c a dãy 2.5.1. Ph ng pháp s d ng đ nh nghĩa tính gi i ươ h n Nh n xét: ta có th d đoán đ c gi i h n c a m t s dãy s nh vi c tìm ượ nghi m c a ph ng trình liên quan. D đoán này c n đ c ki m nghi m l i ươ ượ b ng đ nh nghĩa gi i h n c a dãy s . Ta nh c l i r ng, dãy { x n } đ c g i là h i t n u t n t i m t s a sao cho dãy ượ ế { x n a } là vô cùng bé, nghĩa là v i m i > 0, luôn t n t i s
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